Alex Kontorovich on the Absolute Truth of Pure Math
Alex Kontorovich, professor of mathematics at Rutgers University, speaks with host Steven Strogatz about the intellectual satisfactions of spherical geometry and about regaining his creative freedom during an intimidating collaboration with Jean Bourgain, a giant of modern mathematics. This episode was produced by Camille Petersen. Read more at QuantaMagazine.org. Original music and production by Story Mechanics.
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Transcript
Steven Strogatz: Well, okay, so, all I’m asking is for you to bare your soul.
[LAUGHTER]
Alex Kontorovich: Right. Let’s do it.
Strogatz: Okay, well, so let’s do it, then.
Strogatz (narration): From Quanta Magazine, this is “The Joy of x.” I’m Steve Strogatz. In this episode, Alex Kontorovich.
[MUSIC PLAYING]
Alex Kontorovich delivered a lecture at our weekly colloquium in the math department at Cornell. You’re in this big space with blackboards all around, 360, all the chairs are lined up so maybe, like, 100 people can sit in there. They’re all facing the main long wide blackboard in the front. The ultimate in style is to start with a blank blackboard, pick up a piece of chalk, and start on the far-left and start writing.
Alex keeps turning around as he writes, and he really talks to us, and he makes sure that we’re paying attention and getting it. You get the feeling that he’s enjoying himself in this performance, almost like, you know, when you used to see Leonard Bernstein, or some other great conductor, smiling as they’re waving their baton around conducting the orchestra. So, for Alex, the orchestra is all these mathematical symbols and shapes and numbers, equations, and he’s in the middle of this bravura performance, with math as his instrument. So, mathematicians make a big distinction, unfortunate, in my opinion, between pure math and applied math. And Alex is a consummate pure mathematician, and I am a, an applied mathematician.
Kontorovich: The pure-applied divide. I mean, I would actually, I would guess that you have a better sense of what I do all day than I do of what you do.
Strogatz: Well —
Kontorovich: In other words, that I can imagine —
Strogatz: — you might, you might think that, but I might think the opposite.
[LAUGHTER]
Kontorovich: Yeah.
Strogatz: But go ahead, finish your thought, why, why do you think that I —
Kontorovich: Well, just because kind of, pure math is much closer, I, I think, to — what, you know, it’s some souped-up version of what I had in ninth-grade geometry, right? There was, like, there were problems, and you had to solve them, and you had to write a proof, and it had to be this, you know, rigorous thing with a foundation, and you build on top of it, and at the end you get some, some nice statement. And so, you know, in sciences, there is, like, what, what constitutes proof depends on which science you’re in.
Strogatz: Aha.
Kontorovich: And the thing that I love the most about pure mathematics is that proof — you know, this is how we get out of jury duty:
[LAUGHS]
Kontorovich: Proof, for us, is not beyond a reasonable doubt; it’s beyond any doubt whatsoever for any human being for the rest of time.
Strogatz: Oh [LAUGHS] — wait a second, now, have you actually used this gambit to get out of jury duty?
Kontorovich: I haven’t used it as explicitly as I just said, but I certainly, I was voir dire-d, once, uh, when I was in New York City, and, uh, you know, was facing some god-knows-how-long court thing. And, uh, you know, they asked, “What do you do?” and I said, “I’m a, I’m a pure mathematician.”
Kontorovich: And then, I was very promptly kicked off the jury.
Strogatz: [LAUGHS] I see.
Kontorovich: So —
Strogatz: I see. The reason you think that, um, this might work, if it ever comes to it, is that, for us, or, let’s say, for pure mathematicians, the standard is, as you say, not just beyond any reasonable doubt, but beyond any doubt, period.
Kontorovich: For all time.
Strogatz: For all time.
Kontorovich: Yeah, for all time. Euclid’s theorems are still true today, 2,300 years ago.
Strogatz: Uh-huh?
Kontorovich: And they will be true 2,300 years from now.
Strogatz: That is a kind of distinctive thing about math, isn’t it?
Kontorovich: Yeah. So —
Strogatz: So, you say that’s the thing you love — did I hear you right? You say that’s the thing you love most about pure math?
Kontorovich: Yeah, I think so, yeah, the absolutism of it.
Strogatz: Okay, so that’s the thing that a lot of mathematicians would say that they were proud of, that, uh, it’s, if you ask how is math different from physics or biology or anything else, that, that we have — I keep saying “we” — okay, so I’m in the club a little bit, I mean, I do think of myself —
Kontorovich: Of course, you are in —
Strogatz: — as a mathematician in a subspecies. We, we’re mathematicians who see, uh, the world in our image, kind of, you know, like, —
Kontorovich: Mm-hmm.
Strogatz: — that math is everywhere, we like to say. And it isn’t quite the math that’s in our heads, but it’s a, uh — I don’t know — like, often, an imperfect reflection of the, the perfect structures in our heads.
Kontorovich: Yeah.
Strogatz: And it’s an uncanny thing that nature so often seems mathematical, and so, we wake up in the morning fired up to think about that and discover more of that. Um —
Kontorovich: Yeah, the unreasonable effectiveness.
Strogatz: Yeah, what people call the unreasonable effectiveness of, of mathematics. Unreasonable because —
Kontorovich: Mm-hmm.
Strogatz: — why should a, an idealized fiction, really, in our minds, map on to anything that’s out there in the —
Kontorovich: Mm-hmm.
Strogatz: — nitty-gritty real-world. So, you tell me, how do you think about pure math, if I’ve described it as this inward-outward description, do you have a way of saying it?
Kontorovich: I enjoy learning about the way pure math gets applied. But, um, you know, when I have a “application” to my work, it’s to somebody else’s question that was also asked for no reason other than curiosity’s sake. But, you know, as, uh, I think societies that have invested in that kind of, uh, research and that kind of inquiry have found, time and time again, uh, it has these huge implications to humanity, over time —
Strogatz: Mm-hmm.
Kontorovich: — it, it turns out. And you never know where it’s going to come from. So, for example, since we talked about Euclid a second ago, just the, the stupidest thing of axioms, you know, what axioms are you allowed to have or what axioms are you not allowed to have, like, the, the parallel postulate in, in Euclidian geometry. People spent 2,000 years trying to prove that the, that the parallel postulate could be proved from the first four. It was the fifth of Euclid’s axioms, and people thought that it was, you know, not necessary, and they wanted to prove it. That’s as, you know, if I’m going to be, uh, mean about it, that’s as dumb a question as, as one could encounter.
Strogatz: [LAUGHS] So just to remind anyone who hasn’t thought about this, or never actually read Euclid, you know, he’s a mathematician from around 300 B.C. He’s in Alexandria, in Egypt. He’s the greatest mathematical textbook writer of antiquity, and maybe even of all time, and he’s compiling and adding mathematical ideas that have accumulated over the past several hundred years, from all kinds of different cultures, Greeks, but also Babylonians, Egyptians — he’s putting kind of the world’s knowledge of, of math together. And when he writes down these four, or, uh, as you say, five basic postulates that he’s going to build everything else from, the fifth one has a very different character than the first four. The fifth one looks much more complicated, so much so that you get the feeling that it’s, it’s not — that it should be possible to derive it logically from the first four, that it’s not separate, that it should be a consequence. And that was the big question, right? I mean, why is there this ugly — not ugly, but it seems so much more roundabout.
Kontorovich: That’s right. That’s right, it, it’s, it goes something like: given two lines and a third line crossing the two, if the two angles on opposite sides add up to less than two right angles, then when you extend the two original lines indefinitely, they’ll meet in the plane on that side. I mean, it’s, it’s an entire paragraph. It’s actually comical —
Strogatz: It is comical.
Kontorovich: To, to look at the elements, at some, you know, either the Arabic or the original Greek version, whatever it is, there’s, like, one line for Postulate One, one line for Postulate Two; Three and Four are just, like, little half-sentences. And then there’s an entire paragraph that’s an absolute mouthful that you have to chew on for about a, you know, a good five minutes, to figure out what the hell he’s saying.
[LAUGHS]
Kontorovich: And then, let’s say there were 50 propositions, you know, the actual theorems that he proves in Book 1. The first 25 of them, he does not touch the 5th postulate.
Strogatz: Isn’t that interesting?
Kontorovich: He does everything everything —
Strogatz: Yeah.
Kontorovich: Yeah, that’s the psychology.
Strogatz: I mean, reading between the lines, you can feel — because he doesn’t — he’s a very tight-lipped writer, this guy, Euclid. He doesn’t —
Kontorovich: Yes.
Strogatz: He doesn’t show you his emotions at all, compared to Archimedes —
Kontorovich: Right.
Strogatz: — who is my favorite, who’s all, very emotional, to me, compared —
Kontorovich: Yeah.
Strogatz: Euclid is Mr. Tight-to-the, to-the-Vest, and the fact that he uses these 25 or whatever propositions, before he even deploys the use of the 5th, the parallel postulate, shows you —
Kontorovich: Yeah.
Strogatz: — that he’s kind of squeamish about that one. He wants to do —
Kontorovich: Absolutely.
Strogatz: — as much as he can without invoking this one dubious one, or this one kinda nasty looking one.
Kontorovich: And so, here’s this attempt at proving the parallel postulate, why, why can’t it, at least in theory, be the, the, the case that there is some very clever proof. Then, all you have to do is work hard enough and, and get at it and find it.
Strogatz: Mm-hmm.
Kontorovich: So that’s the kind of thing that I think is quintessential pure mathematics. There’s no reason to be asking this question. This will not enlighten anything in the real-world, in so far as anyone can tell, in asking this question.
Strogatz: Uh-huh. Yes.
Kontorovich: And yet, the act of asking this question forces you to discover spherical and hyperbolic geometry. And if someone just asked this question of me, maybe my younger self, I would say, “Yeah, I don’t know, who really cares? Like, can it be proved? Can’t it? It’s obviously true. Let’s just move along.”
[LAUGHS]
Kontorovich: But when you ask these questions, as a society, maybe, a society — meaning maybe over the span of 2,300 years asking this question, and in, in 1870, or whatever, finally having the answer. And the answer is, no, you can’t, uh, prove the fifth postulate from the other four, because there are other geometries in which the other four are true and the fifth one is false.
Strogatz: So, Euclid’s Postulates describe flat geometry, but that’s not the only kind of geometry out there. There are these other two geometries, curved geometries, hyperbolic, and spherical geometry, and they have different rules. That’s why no one could prove the fifth postulate, because it’s not true in all geometries.
But just by trying to prove that postulate, we discovered new kinds of geometry.
Kontorovich: Let’s talk about spherical geometry.
Strogatz: Okay.
Kontorovich: Because that’s the geometry that, in hindsight, like, we should be smacking our heads over, arguing about this, uh, Euclidian fifth postulate, because we live on the surface on the surface of a sphere, uh, we should’ve thought about this. So, for example, um, train tracks, right, train tracks are like the quintessential parallel line.
Strogatz: Yeah, they are, yep, okay.
Kontorovich: Okay? So you run train tracks, imagine that there aren’t oceans between continents, and so on. You run — so, first thing, the first, before we run a pair of train tracks, let’s run one train track; you have a unicycle train running along earth.
Strogatz: Okay.
Kontorovich: So what does that look like? Or, or maybe a better vision of it is, you get in an airplane, and you lock the steering wheel into, you know, you put it on cruise control and you lock the steering wheel going straight.
Strogatz: All right, yeah.
Kontorovich: So, what will that airplane do? So, for people, if people haven’t thought about this before, uh, you know —
— get your favorite soccer ball or a globe or whatever, and take, uh, a piece of string, and put it between two points, any two points whatsoever on, on your ball, and pull that string taught.
Strogatz: Mm, yeah.
Kontorovich: And what you’ll see, if you kind, kind of follow the string around, uh, trying to keep it taught, is that the shortest path between those two points, uh, if you extend it all the way around, will be a great circle. So if you cut the sphere exactly down the middle, down any middle, then the, uh, boundary of that will be a great circle.
Strogatz: Yeah, I, I just want to throw in one, for anyone who’s ever flown to Europe, I —
Kontorovich: Exactly.
Strogatz: — this is something that, that struck me. I remember the first time when we were flying from New York to Rome — and my dad used to love asking me geography questions, um, so he would say, “Which is farther north, New York City or Rome?”
Kontorovich: Right.
Strogatz: And, you know, if you think about the weather, you think, “Jeez, Rome, it’s Roma, it’s nice, it’s sunny.”
Kontorovich: Yeah.
Strogatz: You think Rome is farther, it’s gotta be farther south.
Kontorovich: It’s obviously further down, right. [LAUGHS]
Strogatz: Yeah, but it’s about the same latitude and —
Kontorovich: Yeah.
Strogatz: And if you fly due east from Rome, or due east from New York, you’ll get to Rome. But in fact, when you watch the way the plane goes, they don’t fly that way; they fly up hugging Nova Scotia, and then, you know, you’re going over, come in through Ireland, and — and I remember, as a kid, thinking, “Oh, that’s so that, you know, in case we lose an engine or two, we could always land in Canada.”
Kontorovich: Yeah, you’re closer to land — I thought the same thing.
Strogatz: That’s not why — you thought that, too? [LAUGHTER]
Kontorovich: Yep.
Strogatz: No, it’s just the thing you said: if you draw the string, if you attach a piece of string on the globe between New York and Rome and pull out the slack —
Kontorovich: Yeah.
Strogatz: — you’ll end up taking the path that the plane actually take, takes. That’s the shortest path, going up over Nova Scotia and then down past Ireland.
Kontorovich: Yeah, so if you want to fly exactly due east, you have to keep your, you know, steering wheel locked at some angle, and that angle gets worse and worse, the higher up, the closer you are to the North Pole, so.
Strogatz: Mm, right, sure, yeah. Yeah, I think people could picture that, if you’re, if you’re trying to go on a line of latitude that’s just a little bit below the North Pole, you’d have to be —
Kontorovich: Yeah.
Strogatz: — curving like crazy to, to stay on that; you couldn’t fly straight. You’d be constantly turning, you know, either to the left or the right, depending which way around the, the pole you’re trying to go.
Kontorovich: Right. So, now let’s imagine, let’s get back to the train tracks, let’s get to the parallel lines. So imagine you’re standing in New York City and you’re facing exactly due north. So if you keep flying on that airplane, you’re going to go over the North Pole, and then back around down the South Pole, and then —
Strogatz: Mm-hmm.
Kontorovich: — back up to New York, right?
Strogatz: Okay.
Kontorovich: Now, imagine your friend is going to be one foot away from you, pointing exactly due north.
Strogatz: Okay, one foot, uh, let’s say, west of you.
Kontorovich: Let’s say west of you.
Strogatz: Yeah, okay.
Kontorovich: Pointing exactly due north.
Strogatz: Uh-huh.
Kontorovich: So here you are, both of you are moving exactly parallel to one another, and you’re going to move on these greatest — great arcs, right, great circles, these geodesics, these shortest, uh, path directions.
Strogatz: Sure.
Kontorovich: And what’s going to happen, as you, as you keep going farther and farther north? You’re actually getting closer and closer together, until, at the North Pole, you come together.
Strogatz: Mm-hmm.
Kontorovich: So, there are no — no human being has ever seen parallel lines.
Strogatz: That’s an interesting statement [LAUGHS], that there really aren’t parallel lines on the surface of our curved earth. We think there are, but that’s only because —
Kontorovich: We think there are.
Strogatz: — we’ve seen a little bit of them.
Kontorovich: Yeah.
Strogatz: If we could see farther —
Kontorovich: It’s all an illusion.
Strogatz: Uh-huh. So you’re saying spherical geometry, which, which, except for the flat earth people, those of us —
Kontorovich: Right. [LAUGHS]
Strogatz: — that do believe in the round earth, um, would say that, again, Euclid is using this interesting creative fiction, that there are parallel lines. Because in real-life, on the surface of the earth, there really aren’t.
Kontorovich: That’s right.
Strogatz: We’re, we’re drinking each other’s bathwater, here, right, we’re a couple of mathematicians.
Kontorovich: Yeah. [LAUGHS]
Strogatz: And, you know, that same kid who says to you, “I don’t care about trigonometry,” and then you say, “Well, if you want to know the, the circumference of the earth you do,” but that kid says, “I don’t want to know that, either.” [LAUGHS] Why —
Kontorovich: Right.
Strogatz: You know, what’s your — how do you address that? You are a professor, so you have this — occasionally, this comes up, and you have little kids, so you maybe — that, that may come up with them.
Kontorovich: Right. Yeah, yeah, I’m exactly getting to… You know, it’s funny, this, uh, partial solar eclipse, last year, turned me into a, a bit of an amateur, uh, astronomer, because I just started thinking all these questions. You know, they tell, they tell us — “they,” whoever “they” are — they tell us how far away the sun is, and, and how many times bigger it is than, than the earth. But, like, what are the actual — I guess this is the mathematician in me coming out — why? How do you know? Somebody figured it out. And I guess most people are content with just knowing these, these facts, but, uh, the mathematician in me says, “No, I don’t trust anyone else. I don’t trust — “ And, and it’s not a matter of trust, uh, let’s not go flat earth, uh, uh, here. It’s that I want to know, from basic principles, from myself.
Strogatz: Yeah.
Kontorovich: That’s what proof allows. Proof is a kind of — mathematical proof is like antiauthority.
Strogatz: That’s so interesting.
Kontorovich: There is no one — yeah.
Strogatz: Okay, go ahead, finish that thought. It’s so interesting, because a few, you know, minutes ago, you said what you loved about math is the absolutism of it, the absolute certainty.
Kontorovich: Yes.
Strogatz: Which might, uh, bother some people, makes them feel like math is the most authoritarian of all subjects. And yet, you’re saying, now, “No, no, no, it’s exactly the opposite. It’s the, the subject where —
Kontorovich: Well —
Strogatz: — “if you think for yourself —
Kontorovich: Yeah.
Strogatz: — “that’s all you need.”
Kontorovich: So it’s both.
Strogatz: It’s both.
Kontorovich: It’s both. You should wrestle with it, but it will defeat you [LAUGHS], because, at the end of the day, uh, as long as you’re following, you know, the logic that, uh, everyone else assume — you know, we, we sort of, we have these logical principles that we adhere to. And if you follow those logical principles, they will wrestle you into submission. There is no other option for how many, you know, the angles in a triangle, in, in Euclidian geometry, once you allow yourself that, that parallel postulate. You, it’s completely free for you to think and argue and decide what questions you find interesting, and try to measure the circumference of the earth for yourself. But once you measure it, it’s an absolute; there’s an answer, there’s an absolute certain —
Strogatz: So, let, let’s lay this down. So you’re saying there’s, there’s the freedom in the concepts and in the definitions and in the axioms, but not in the answers.
Kontorovich: Right.
Strogatz: Is, is that it?
Kontorovich: It’s, it’s the age-old, “Is mathematics, uh, an, an invention or a discovery?”
Strogatz: Yeah, what do you think?
Kontorovich: So, the questions that are being asked are an invention; the answers are a discovery.
[MUSIC PLAYING]
Strogatz: After the break, why Alex is like a hockey player born in January, what his career in math means to his mom, and what it was like to be mentored by a modern mathematical giant.
[MUSIC PLAYING]
Strogatz: I know you do jazz music yourself; you’re a great saxophonist and clarinetist. Now that you’re good at clarinet and saxophone, I assume you do it partly because you like your craft, you like what you can do. And so, is, is what you like about math that you’re good at math and you like doing your craft, as opposed to this, this lofty thing about the philosophy of absolute truth?
Kontorovich: Yeah, that’s a, that’s a really good question. Growing up, I mean, I came to this country as an immigrant, uh, I didn’t speak a word of English, so when I got to school, they handed all this, uh, you know, stuff with letters on it, and I didn’t know what any of it meant. And then they handed some numbers and, and symbols, and I was, like, “Oh, yeah, that stuff, we did last year.”
Strogatz: You were eight years old, right?
Kontorovich: That’s right. Uh —
Strogatz: Okay, and, and why don’t you just tell us a little about that. So you are from — where?
Kontorovich: I was born in Russia.
Strogatz: Uh-huh?
Kontorovich: And, uh, came to the US at age eight, and started third grade.
Strogatz: And so, you couldn’t speak or read English, but you knew a little bit about numbers, which were written the same way in Russia, I guess.
Kontorovich: Exactly, the math was the same and was, was about, I don’t know, a year or so behind what, what we had been covering.
Strogatz: Oh, okay, so you were ahead in math, then.
Kontorovich: Well, that’s the point, it’s kind of like the, the Malcom Gladwell, uh, the NF — the NHL hockey players are all born in January or February.
[LAUGHS]
Kontorovich: So, for me it’s the opposite, it’s that, when I arrived in the U.S., I didn’t speak a word of English, and the only subject that I could excel in was the one that had been going much slower here than, than it had been, uh, in Russia.
Strogatz: And can I just ask a little more? So, your mom is a mathematician. Does that mean a math teacher, a research mathematician? What, what exactly?
Kontorovich: Yeah, so, this is, uh, I mean, it’s a bit of a long story, but she was, um, she had written a Ph.D. dissertation, and, uh, was not allowed to defend it, as, you know, there’s the standard kind of Jews in the Soviet Union thing.
Strogatz: Okay.
Kontorovich: So, um —
Strogatz: So, Jewish family —
Kontorovich: So, she —
Strogatz: She, she couldn’t get a doctorate, because of the, the antisemitism in former Soviet Union.
Kontorovich: Right.
Strogatz: Uh-huh? Uh —
Kontorovich: And so, that’s part of the reason we left, and when we arrived here, she, um — she actually, uh, for a, a, a hot minute, I think, uh, attended Courant. She sort of, she, she was admitted to, to their program, and she said, “Okay, here’s my dissertation. I’d like to graduate and, uh, apply for jobs now.” [LAUGHS] And they said, “Wait, that’s not how it works, you know, you have to have some exams, and take some courses and stuff.” And she had a family to provide for, so she went into finance.
Strogatz: Mm, but, so, okay, just to fill in, Courant refers to the Courant Institute —
Kontorovich: Yes.
Strogatz: — one of the great math institutes at NYU and in, in New York, obviously — [LAUGHS]
Kontorovich: Yes.
Strogatz: — one of the great ones in the world. Okay, but then she went into finance, you say.
Kontorovich: Right.
Strogatz: Uh-huh.
Kontorovich: And so, she’s made her career in finance, and it wasn’t until — it’s actually funny, because, you know, I mean, I always knew she was a mathematician, uh, in the sense of someone who —
Well, she, she did prove original theorems, at some point in her life, but certainly by the time I realized what that word meant, she wasn’t doing that anymore. But, um, when I, when I defended my thesis, you know, you, they, they — you have to make a copy of your, uh, thesis for your advisor, and one for the library, and I made an extra one just to give to my folks. So I gave it to my mom, um, and she got a funny look on her face, and she went to her files and she, like, dug around, and she pulled out her thesis —
Strogatz: Yeah?
Kontorovich: — that had never been published. And she gave it to me, and it’s on Navier-Stokes.
Strogatz: Get outta here.
Kontorovich: And — yeah, so —
Strogatz: But, wait, so, wait —
Kontorovich: — I still have it —
Strogatz: But, wait, why was she having a funny expression on her face?
Kontorovich: I guess she was reminiscing about the time that she was hoping to defend her thesis, and here she was —
Strogatz: I see — I thought you were going to tell me she wrote her thesis about the same topic, but that didn’t happen.
Kontorovich: No, no, yeah, that would be —
Strogatz: Not quite like that, but okay.
Kontorovich: Not quite like that, but the, the feeling of, you know, she — this is the land of opportunity. We left the Soviet Union, and she brought us here where we could defend our Ph.D. theses [LAUGHS], if we so chose.
Strogatz: Yeah. So your mom was a mathematician, then went into finance. You yourself were a couple years ahead of the other kids, in elementary school, when you got here, in math.
Kontorovich: Right.
Strogatz: Okay.
Kontorovich: Right. So, math kind of always came easily, and that’s what I mean by I was kind of like a January baby in Canadian hockey league.
Strogatz: I have to admit that Alex’s talent intimidates me. Some of my colleagues who were the top mathematicians in the world themselves have told me that Alex is at the very top.
Strogatz: You’ve done amazing stuff in math, but where I wanted to go, right at the moment, was this feeling of being surrounded by people at the different levels — you know, tenure track professors, and people with tenure, and people who are recognized as the elite of the elite, and so on. Here’s what I’m sort of thinking in my dark moments. I think, “Why bother?” You know, like, why don’t I just leave it to these people who are so much better at this than I am, so much smarter, so much faster, so much more knowledgeable —
Kontorovich: Hm —
Strogatz: Do, do you ever have thoughts like that, and do you answer your — do — what’s your answer to yourself?
Kontorovich: Yeah, I answer myself very quickly.
Strogatz: What do you say?
Kontorovich: That’s a thought that I have all the time, and I answer myself very quickly: I love it. If I don’t, if there’s —
Strogatz: Wait a second, you do have this thought all the time, this self-doubt —
Kontorovich: I do have this thought — yeah, I mean, I, I don’t know that I have the thought, like, um, “I’m not cut out for it. Lemme quit.”
Strogatz: Yeah.
Kontorovich: I definitely have the thought, “Holy crap, that guy is so much better,” or, “That girl is so much better than I am.”
Strogatz: [LAUGHS] Yeah, yeah.
Kontorovich: Uh, but that’s okay because I still enjoy it.
Strogatz: Nice.
Kontorovich: And if, you know, if someone wasn’t paying me, I would probably be just be sitting in my, in my house doing exactly the same thing that I’m doing except getting paid, so.
Strogatz: [LAUGHS] Well, that’s beautiful.
Kontorovich: I don’t know if, I don’t know if that’s an answer.
Strogatz: Well, okay, but, so when you get to work with these phenomenal talented mathematicians —
Kontorovich: Yeah.
Strogatz: — rather than feeling psyched out about, you know, some kind of inferiority feeling, uh, it sounds like you just delight in their company.
Kontorovich: Yes. Although, to be fair, I should say, I mean, I, I’m not, um, thrilled with this story, but since I told it at Bourgain’s, uh, birthday conference, a couple years ago, which was apparently recorded, I guess I can repeat it here.
So, this is Jean Bourgain, who, uh, is a Fields medalist and — he, he’s basically a modern, modern-day Archimedes, one, one can argue.
Strogatz: Wow.
Kontorovich: He’s really at the absolute highest level and at his — he just passed away, and, um, there was a memorial conference for him at the Institute for Advanced Study in Princeton. And Fields medalists were saying, “I couldn’t believe how good this guy was.”
Strogatz: Wow.
Kontorovich: So, uh —
Strogatz: So this is like saying the all-star team of the NBA talking about —
Kontorovich: Yes.
Strogatz: — another player as, like, “I can’t believe how good that player is.”
Kontorovich: Yes. It’s, like, yeah, Magic Johnson and Bird saying, “Man, Jordan is so good.”
Strogatz: Yeah.
Kontorovich: You know, he’s, he’s that, so, Bourgain is that —
Strogatz: You’re showing, you’re showing your age, there.
[LAUGHS]
Strogatz: That is not what anyone would say today — who should we be saying? We should be saying Steph Curry, Kawhi —
Kontorovich: Right.
Strogatz: — and LeBron? I mean, who’s, who would you say today?
Kontorovich: LeBron, yeah, exactly.
Strogatz: Okay, but anyway, so you’re at Bourgain’s memorial service —
Kontorovich: Right.
Strogatz: — and this is the Michael Jordan of, of, uh, math.
Kontorovich: Right, and, and people — who were — exactly — people who were themselves, you know, the elite of the elite are saying how he was the even more elite than, than them, I mean —
Strogatz: Mm —
Kontorovich: And this isn’t just, uh, this is stuff that I heard when he was alive, too, so it’s not just people being nice at someone’s —
Strogatz: Okay, uh-huh.
Kontorovich: — uh, memorial.
Strogatz: Wow.
Kontorovich: Um, so when I, when I, when we first, I mean, we — so he and I wrote, I don’t know, a dozen papers together, or something like that. And when we first started working together and I would go around and give, you know, talks on what we were doing, after the talk or at dinner or at tea or whatever, someone would pull me aside and say, uh, “So, you, you’re really working with Bourgain. Like, what, what is that like?”
Strogatz: [LAUGHS] Yeah.
Kontorovich: And I never understood what they were saying, because… So, apparently, when he was younger, he was very kind of — I don’t think he was ever not nice, he just, once he understood what your theorem was — which was usually extremely quickly, and maybe he had an even better proof of it and had a more general, uh, could prove a more general theorem — uh, he maybe would decide he’s no longer interested in the conversation and go, you know, uh, do something else.
Strogatz: Yeah. [LAUGHS]
Kontorovich: Um —
Strogatz: Yeah?
Kontorovich: Uh, which, you know, is purely hearsay. I never saw anything like this, I never witnessed anything other than, you know, him being a complete sweetheart and, you know, gentle and supportive of everyone around him, and just trying to lift everybody up.
Strogatz: Mm —
Kontorovich: Um, but that said, when we first started working together, I have to admit that I was, uh, petrified, you know, uh, just being at the blackboard with this guy whose reputation preceded him and is this absolute giant. And he’s saying stuff on the blackboard, I have no idea what he’s saying, I’m just, you know, jotting down in my notebook, as fast as I can. And, um, at the end of our — you know, we would work for six hours or something, and I’d go home and spend the next three weeks trying to figure out what the hell it was that he was talking about.
Strogatz: [LAUGHS]
Kontorovich: And, uh, once I’ve written out 50 pages of calculation and eventually make sense of, of what it is, then I’d come back and, and, you know, we’d, we’d talk again. But I remember, in the very early part of our collaboration, we would be at the blackboard.
I would have some idea, I would want to say, “Well, wait a second, can’t we take whatever this — Cauchy-Schwarz here, or something, uh, can’t, can’t we, can’t we do this move, there?” Um, and I would, and I would immediately, in my mind, start, like, poopooing the idea and, and finding holes in it, and, and thinking, “Yeah, it’s not going to work because of this, this, and this. Will it really work? I don’t know, I’m not so sure.” And then, you know, five minutes later he’ll say, “Well, why don’t we do Cauchy-Schwarz, here?” and then take off from there.
Strogatz: [LAUGHS]
Kontorovich: And I’ll be kicking myself, “Ah, god, I should’ve said it. Why didn’t I — what, what’s with me? I’m not like this, I’m not like —
Strogatz: Yeah.
Kontorovich: — “I don’t clam up. Uh, I’m, you know, a pretty…” And, and I think it’s my noncompetitive — well, I would like to think that, because I’m not competitive, uh, I’m perfectly happy to make a mistake. I’m not, you know, I’m not, um, cautious — like, this is something about math, you know, mathematicians don’t like to make a conjecture, in case it’s wrong.
Strogatz: Yeah.
Kontorovich: Well, if you make a conjecture and someone else finds out that it’s wrong, or you find out that it’s wrong, you’ve learned something.
Strogatz: Of course.
Kontorovich: Isn’t that great, isn’t that beautiful, you asked an interesting question? But, um, I guess if you have more of an ego about your conjectures need to be correct, then you would refrain from it. So I never felt that way, I was always perfectly happy to make mistakes, and somehow, with Bourgain at the blackboard, I all of a sudden wasn’t perfectly happy to make mistakes. And, um, I confided in Peter Sarnak, uh, who was my Ph.D. thesis advisor, one of my, uh, two advisors, and, uh, Dorian Goldfeld being the other one, and, and I confided in him and said — because he was good friends with, with Bourgain. And I said, “You know, I don’t know what’s going on, when I’m, when I’m working with him, I kind of, um, I’m, like, self-censoring.” And, uh, Peter thought for a minute, and, as usual, immediately diagnosed both the, the problem and the solution. And he said, “Well, you’re worried that you’re not as good as him.”
Strogatz: [LAUGHS] I can imagine what’s coming next. [LAUGHS]
Kontorovich: “You’re not. Nobody is. Get over it.” And that was all I needed to hear.
Strogatz: Wow.
Kontorovich: The second I heard that, I was, like, “Oh, right, good point.” And then, and then the collaboration flourished, and I could say whatever stupid idea I had and — you know.
Strogatz: That’s so interesting. It’s such a relief, right?
Kontorovich: Yes.
Strogatz: “Of course you suck,” you know?
Kontorovich: Right.
Strogatz: That’s … so what?
Kontorovich: Right. Right, “Look at who you’re standing next to.”
Strogatz: Yeah.
Kontorovich: “Of course you suck. Don’t worry about it.”
Strogatz: So, when Alex first applied to work with Bourgain, at Princeton, he got the sense he wasn’t going to be accepted.
Kontorovich: This was, like, a year out of grad school. I applied to the Institute for a yearlong membership; there was, like, a special year in analytic number theory. So, all the superstars in my field were going to be there, and of course I wanted to be there, so I applied. And in my application, I said that I wanted to, um, let’s not get technical, I wanted to do XYZ. I had a proposal to do XYZ, which was going to be kind of, you know, what I wanted to do for the next year, and I thought it would lead to some good results. And I visited Peter Sarnak, to, um, talk about all kinds of various things, and after our meeting, you know, we had worked together for three hours or so, and then, it was teatime and he said, “By the way, Bourgain wants to speak with you.” I said, “Oh? Okay?”
So, so we went to tea, and eventually he showed up at tea, and he saw me and he said, “Ah, okay, come with me,” and he sort of walked out of the, the tearoom, and headed straight for his office. So, I, you know, put down my teacup and [LAUGHTER] dutifully followed him, sort of like going to the principal’s office or something — I’m being summoned.
Strogatz: Yeah?
Kontorovich: Um, and we got to the blackboard, and he just went off into, you know, a five-hour lecture on something. I had no idea what it was. I wrote down — as I said, I just wrote it all down and, uh, over the next few weeks, figured out that he was explaining to me how to do XYZ, the thing that I had proposed in my application —
Strogatz: Oh —
Kontorovich: — to do, that would lead to, you know, hopefully, good things.
Strogatz: Yeah?
Kontorovich: And so, in my mind I was, like, “Oh, well, crap, so, I guess I’m not getting this membership. I guess they’re not going to, you know, they’re going to decline my application, since he just explained to me how to do the thing that I wanted to do.”
Strogatz: Well, it sounds like he just did a year of work for you.
Kontorovich: Right, so I, so I said to Sarnak, uh, you know, “I guess, uh, I guess I’m not getting in.” [LAUGHS] And he said, “No, no, it means he wants to work with you.”
Strogatz: [LAUGHS]
Kontorovich: So that was —
Strogatz: What was — was your XYZ question something that he had not thought of before?
Kontorovich: So, I never talked to him about that, but, uh, after he passed, I was talking to, to Peter, and he said, yeah, that’s something that he would — you know, he would read applications, and he would realize that there’s some, someone’s asking something that he knows how to answer, and he’ll write to them and they’ll start a collaboration.
Strogatz: Well, that’s something that a young mathematician often doesn’t know, that, that a more seasoned mathematician will realize, “Oh, this is going to be beautiful,” but, but in the future tense. That is, you can see, at the beginning, if you’re, if you’re a person like Bourgain, that there’s gold at the — there’s gold in them thar hills. Without — you don’t have to actually see the gold; you just know it’s there. And that we’re going to begin something magnificent, we’re going to begin a fantastic journey together, and it’s going to work.
After the break, the geometry of wineglasses and champagne flutes, and chasing down a mathematical deer.
[MUSIC PLAYING]
Strogatz: What’s the gist of what you did?
Kontorovich: So, I, I, sometimes I try to explain it as, like, imagine three wineglasses.
Strogatz: Okay?
Kontorovich: And they’re all, you know, arranged, packed together, so that they’re touching.
Strogatz: Three wineglasses — so I’ve got ’em standing on the table —
Kontorovich: Yep.
Strogatz: — they’ve got their circular bottoms —
Kontorovich: Let’s make the bottoms so big that the bottoms touch before the goblets do. So, here’s a, here’s a nice computation, here’s, uh, an exercise for your listeners, if they want to do a little bit of high school geometry. Let’s say you have a three-inch wineglass, and you have three of ’em, and you put ’em together so that the bottoms are all touching.
Strogatz: Yeah.
Kontorovich: Okay? And, uh, let’s say you’re also going to serve champagne flutes —
Strogatz: Let me make sure I have the picture, though. So when you say the three are all touching, they’re sort of packed together like I’m trying to pack ’em tightly in my cabinet, and —
Kontorovich: Exactly.
Strogatz: I’m not hanging ’em up on one of those fancy things that lets them ventilate. They’re just, they’re just standing upright, they’re in the cabinet —
Kontorovich: Yeah.
Strogatz: — and their bottoms are all touching in a way that they’re so packed together that they almost make, like, a little triangle.
Kontorovich: Exactly, they make a curvilinear triangle.
Strogatz: Yeah.
Kontorovich: Right.
Strogatz: Okay.
Kontorovich: I think of, like, a, a dinner party, and there’s a tray, there’s, like, a waiter holding a tray, and that tray has the glasses packed very, very tightly —
Strogatz: Okay, yeah.
Kontorovich: — as tightly as possible.
Strogatz: Yeah, yeah, yeah.
Kontorovich: Okay? But you’re not only serving red wine, you’re also serving champagne.
Strogatz: Oh, yeah.
Kontorovich: So you have this option of the two. And the champagne flute is thin —
Strogatz: Okay.
Kontorovich: — and you want to order a champagne flute so that it just perfectly slips in that curvilinear triangle, okay?
Strogatz: Uh-huh.
Kontorovich: So, so, here’s the calculation: what’s the, uh, diameter of that, if the diameters of the wineglasses are three inches?
Strogatz: Aha.
Kontorovich: So, in fact, it’s Apollonius of Perga, uh, another ancient Greek from 250 B.C., who, um, who thought about this question.
Strogatz: Mm-hmm.
Kontorovich: And Apollonius is the one that gives the algorithm for that construction.
Strogatz: So, those three circles could be any size.
Kontorovich: Could be any size, that’s right, there’s —
Strogatz: I mean, I don’t know that any — by now, I, I imagine it’s clear that we’re not talking about real-world utility, that this is —
Kontorovich: That’s right.
Strogatz: — for pure curiosity, you just want to understand things about circles and the way they pack together, for the fun of it.
Kontorovich: Right. That’s right. And —
Strogatz: And we can — I, I’m okay with that, but that’s the kind of thing mathematicians like to do.
Kontorovich: Yeah, and then, you know, when you extend it to higher dimensions, it starts, you start thinking about bubbles and the formation of soap film and, and bubbling in general, and minimal surfaces, and there’s a lot of kind of — there is — it, it turns into, eventually, something that isn’t necessarily just pure thought. In fact, so, what we haven’t gotten to, at all, is the, the arithmetic properties of this thing.
Strogatz: Yeah.
Kontorovich: Which, um, so, maybe we can talk about that for a second.
Strogatz: Let’s go ahead, yeah.
Kontorovich: Okay. So, so, now, uh, we’re going to change it yet again, but, but keeping the same metaphor. Let’s say you have a circle of radius a half, let’s say half an inch, and another circle of radius half an inch, and you put those two together. You, you kind of, you see that?
Strogatz: Yeah, just touching the same way we’re, we’ve been doing.
Kontorovich: Just, yep, just two circles touching —
Strogatz: Yeah.
Kontorovich: — radius half an inch. So, their diameter’s an inch —
Strogatz: Yeah.
Kontorovich: — which means, the point where they touch, there’s a circle of radius one —
Strogatz: Mm-hmm.
Kontorovich: — that kind of perfectly encloses those two.
Strogatz: Yes.
Kontorovich: Okay, and that creates these two, let’s say you have them north-south, so that creates two curvilinear triangles, an east one and a west one, which are identical, I mean, across the —
Strogatz: So, you’re saying: Put, put one penny, and then another penny on top of it, they’re both —
Kontorovich: Right.
Strogatz: — on the table, and now draw the biggest circle — not the biggest — draw a big circle around both of them, that’s just touching at the north pole and the south pole.
Kontorovich: Exactly.
Strogatz: Yeah.
Kontorovich: So you’ll have two of these curvilinear triangles on the east side and the west side —
Strogatz: Mm-hmm.
Kontorovich: — and what is the circle that — what’s the, uh, diameter of the circle that perfectly sits inside there? And it turns out the — so I’ll give away the answer to this one.
Strogatz: Okay.
Kontorovich: It turns out the answer is a third.
Strogatz: That’s the, uh —
Kontorovich: Okay?
Strogatz: — the radius, you say, one-third. If the others were radius a half —
Kontorovich: Yeah.
Strogatz: Then —
Kontorovich: If the others were radius a half, then this one has radius a third.
Strogatz: And that’s, and that’s something where we should be whooping it up, right, we should be rejoicing, because why are the numbers two and three — and actually, there’s also the number one, because the one —
Kontorovich: Yes.
Strogatz: — is the diameter of the big circle that surrounds the whole picture.
Kontorovich: Yes.
Strogatz: Somehow, the numbers one, two, three, which are the way that every child learns to count, they’re magically appearing in this question about packing circles together. And that seems —
Kontorovich: Yep.
Strogatz: — to us in math, that’s, that’s ecstasy, that’s euphoria, right? That the two worlds of numbers and pictures have suddenly collided, numbers, uh, geometry and arithmetic have suddenly come together.
Kontorovich: Yeah. Yeah, it seems unlikely, given, I mean, if you actually follow Apollonius’ construction, all the different things you have to do to get this next circle, you know, it’s highly nontrivial to draw that next circle. And so, you would think, you know, where’s the pi, or where’s the root two, or where’s, you know, —
[LAUGHS]
Kontorovich: There’s no reason for that to be a rational number.
Strogatz: That’s —
Kontorovich: Not only is it a rational number, it’s a, it’s one over an integer.
Strogatz: Yeah, one over one, one over two, and one over three.
Kontorovich: So if you start iterating this — and actually Leibniz was the first person to, as far as we know, there’s notebooks of Leibniz where he starts iterating this Apollonian construction, and drawing more circles everywhere he sees a curvilinear triangle.
Strogatz: Mm-hmm. Maybe you should say, what is it, when you, when you keep talking about curvilinear triangle, can you say that in plainer words?
Kontorovich: Yes. So, let’s go back to the two circles of, uh, radius a half —
Strogatz: Yeah.
Kontorovich: — and the one circle of radius one.
Strogatz: Yeah.
Kontorovich: That’s kinda the easiest thing to see. So, you have these two, you have a north-south circle, and then you have a circle that contains both of them.
Strogatz: Yeah.
Kontorovich: So, there’s the space between, uh, the, the three, the three circles —
Strogatz: Mm-hmm.
Kontorovich: — that little area, and it’s a triangle in the sense that it has three sides. But those three sides are not straight lines, they’re, um, arcs of a circle.
Strogatz: That’s it. When someone hears “triangle,” if they think with straight lines for the three sides, they’re going to have the wrong picture.
Kontorovich: Right, right.
Strogatz: This is a triangle where the three sides are themselves circular arcs. So it looks a little bit —
Kontorovich: Exactly.
Strogatz: — like a claw that I use to comb my dog, that —
Kontorovich: Yes.
Strogatz: — there’s a thing with a circular top, and then it’s got two circular — it’s got, it’s got, like, three teeth that I can use —
Kontorovich: Exactly.
Strogatz: And, and those teeth are formed by two circles touching in, in the right way, two arcs of circles, half-circles, actually.
Kontorovich: Exactly.
Strogatz: Yeah, okay, but anyway, so this, this funny claw-shaped thing with the three fingers is, is your curvilinear triangle.
Kontorovich: So, notice that when you put that circle of radius a third in —
Strogatz: Yeah?
Kontorovich: — you create three more claws —
Strogatz: Oh, yeah. Wow.
Kontorovich: — from the holes that it leaves behind.
Strogatz: Uh-huh?
Kontorovich: And then if you put another circle in one of those claws, you create three more claws.
Strogatz: Oh, boy. [LAUGHS]
Kontorovich: And you just keep doing this forever.
Strogatz: And they’re getting smaller and smaller.
Kontorovich: And they’re getting smaller and smaller. So the next one that you put in will have radius one-sixth —
Strogatz: Mm —
Kontorovich: — which, again, is a, a bit of a miracle. Why is it the reciprocal of an integer, one over an integer?
Strogatz: Well, at this point —
Kontorovich: And the next one you put in, you’ll have radius 1/14.
Strogatz: Really? That I wouldn’t see coming. Because I think a lot of people might’ve tried to guess, if you’ve told me that the numbers, so far, have been one over one, one over two, and one over three, and then you told us the next one was one over six —
Kontorovich: Right.
Strogatz: — it seems like I would’ve never guessed 14 is the next one in the denominator. Because I’m thinking the wrong pattern.
Kontorovich: Right, right. The pattern is, uh, rather complicated, and in fact, that pattern is literally the biggest open problem in —
Strogatz: Oh.
Kontorovich: — uh, thin groups. Radius a half, radius a half, radius one, configuration of circles. Then you have this next circle which has radius of third.
Strogatz: Mm-hmm.
Kontorovich: And the next one you put in is radius of six. And, and so on, 1/14, whatever the, whatever the next ones are. So they’re all one over an integer, okay? Here’s the basic question: which integers will you see? For some people, they’ll hear that and they’ll go, “I don’t care.”
Strogatz: Sure.
Kontorovich: “That’s not a question I’m interested in.”
Strogatz: Right, sure.
Kontorovich: Uh, which is perfectly valid. Uh, for people that do care, so, we still don’t know, we still don’t — we have a conjecture that’s called the local global conjecture, because it turns out that there are some numbers, uh, which we can guarantee will not be there. Okay, you give me, uh, 1,000,024 or something —
Strogatz: Yeah.
Kontorovich: — I don’t know, and I say, okay, I don’t know — I don’t know if there’s going to be a circle with that radius, but I’m going to check a couple of these residues, a, a couple of the remainders, I’m going to divide your number by a couple of numbers, and check what those remainders are. And if it comes out — if the answer comes out wrong, I can immediately tell you that that number will not be one of the circles.
Strogatz: Mm-hmm.
Strogatz: Here’s what Alex is saying: we have these circles stuck between the wineglasses, that are getting smaller and smaller and smaller, and we’re trying to predict the size of their radii, but we can’t. What we can do is eliminate some of the options, by doing certain division problems and then looking at the remainders that come out.
Kontorovich: So, it’s easy to show what’s disqualified, and then, if a number isn’t disqualified, well, then the conjecture is, as long as it’s, you know, beyond a certain point, as long as the number is sufficiently large, uh, it should actually arise as the radius, or one over the radius, of some circle in the packing.
Strogatz: Mm, so it’s something like that, that whatever is not disqualified by what you’re calling the local obstructions, these, these criteria where you could look at remainders and say, “Ah, you’re out,” um —
Kontorovich: Yeah.
Strogatz: — whatever is not forbidden on those grounds is required in the sense that it has to occur — so this is this local global thing that you’re talking about —
Kontorovich: That’s the conjecture —
Strogatz: — about the circle.
Kontorovich: — and I can tell you the state-of-the-art theorem —
Strogatz: Yeah.
Kontorovich: — in, in one line.
Strogatz: Yeah?
Kontorovich: It’s that the conjecture is true in the sense of density, in the sense of probability. So in other words, if we look at how many numbers, out to a million, that should be there and are there —
Strogatz: Yeah?
Kontorovich: — if we look at, like, let’s say, the proportion —
Strogatz: Yeah, yeah.
Kontorovich: — of numbers that do arise versus the numbers that should arise.
Strogatz: Mm-hmm.
Kontorovich: We look out to a million or we look out to a billion or out to a trillion, and we take that, that limit out to infinity, that ratio converges to one.
Strogatz: You say that’s now known to be true.
Kontorovich: That is now known to be true, right. So that’s a —
Strogatz: And is that something that you did with one of your people?
Kontorovich: With Bourgain, yeah.
Strogatz: That’s what you and Bourgain, one of your 12 papers showed this statement.
Kontorovich: That’s right.
Strogatz: That the statement becomes almost — well, what’s the language? — it’s like it’s almost certainly true, as we go out to larger -?
Kontorovich: It’s almost certainly true.
Strogatz: Yeah.
Kontorovich: If you pick a huge number at random, and, uh, you know, make sure it’s not ruled out by these, uh, trivial — it, it passes these, these trivial checks —
Strogatz: Yeah.
Kontorovich: — then, almost certainly, I mean, asymptotically, almost certainly, uh, it does, it does arise as, as the reciprocal of a circle.
Strogatz: You’re not claiming that it’s simply true, you don’t have that, yet.
Kontorovich: Sure, yeah.
Strogatz: It’s, there’s a difference, right?
Kontorovich: There’s a big difference, yeah.
Strogatz: Tell us about that.
Kontorovich: I mean, well, we, we — I sort of stated it in this probabilistic, uh, interpretation.
Strogatz: Yeah.
Kontorovich: But the, the, the theorem is there’s this ratio, and that ratio goes to one.
Strogatz: Mm-hmm.
Kontorovich: So, that’s it. But now, you could have, let’s say, you know, just look, look in the numbers, almost all numbers are not squares.
Strogatz: Mm-hmm.
Kontorovich: A hundred percent of numbers are not square numbers.
[LAUGHS]
Kontorovich: You look at the fraction of square numbers relative to the fraction of all numbers, up to a million or a billion or a trillion.
Strogatz: Uh-huh?
Kontorovich: You take that, that ratio, that ratio has a limit and that limit is zero. So a — none of the numbers are squares.
Strogatz: Well, yeah, so let’s just make sure that that’s completely obvious, what, what you just said.
Kontorovich: Please.
Strogatz: So, like, of all numbers less than 100, there are only something like 10 numbers that are perfect squares, right?
Kontorovich: Exactly —
Strogatz: One squared, two squared, three squared —
Kontorovich: — because it’s square root of — yeah.
Strogatz: — you know, nine, one, four, nine.
Kontorovich: Yeah.
Strogatz: So, so only 10 out of 100 are perfect squares —
Kontorovich: Right.
Strogatz: — so only 10 percent of them.
Kontorovich: Right.
Strogatz: And, and in general, as you say, you take the square root of the big number, to figure out how many perfect squares there are.
Kontorovich: Right, so 1,000 of the numbers, up to a million, are squares.
Strogatz: Ah, okay, right.
Kontorovich: Because 1,000’s the square root of a million.
Strogatz: Cool. And so, if we’re doing a trillion — that’s a number we hear about a lot with the debt — trillion would be 12 zeros.
Kontorovich: Right.
Strogatz: And so, the square root of that would be something with six zeroes, which we call a million.
Kontorovich: Six zeros, right.
Strogatz: So, 1/1,000,000 of the numbers less than a trillion are perfect squares.
Kontorovich: Are perfect squares, right.
Strogatz: So, 1/1,000,000 is kind of almost nothing, but it’s not nothing —
Kontorovich: Right.
Strogatz: — but it’s almost nothing.
Kontorovich: Right.
Strogatz: That’s what you’re talking about.
Kontorovich: But in the limit, it goes to nothing.
Strogatz: Right.
Kontorovich: So the statement would say almost no numbers are squares.
Strogatz: Yeah. [LAUGHS]
Kontorovich: But, of course, there are squares, many of them.
Strogatz: There are squares, but they’re kind of hard to hit. You gotta get very lucky.
Kontorovich: Yeah.
Strogatz: Your, your —
Kontorovich: Yeah.
Strogatz: — idea might be that there are an infinite number of counterexamples, but they’re very sparse.
Kontorovich: Yeah. Our theorem cannot rule out an infinite number of counterexamples. But —
Strogatz: But in your heart, what do you believe?
Kontorovich: Well, I believe the conjecture, and the conjecture says there should only be finite numbers of them.
Strogatz: But there’s only finitely many bad boys.
Kontorovich: Yep.
Strogatz: Okay. This is off the topic a little, but help, help me picture the day-to-day existence of, of someone doing what you do.
Kontorovich: Yeah, it depends, so, it’s a lot of, you know, papers, printing papers, and sitting there with the paper and pen, and trying to understand what someone’s written, doing calculations on, on paper.
Strogatz: You mentioned, earlier, that when your collaborator, Bourgain, was, was giving an exposition at the blackboard for five hours, that you were taking notes. I don’t —
Kontorovich: Uh-huh.
Strogatz: I don’t think I’ve ever taken notes on any, anything that any collaborator of mine said. I, I have, I have worked with people where they wrote on a piece of paper —
Kontorovich: Right.
Strogatz: Uh, maybe that’s the thing, because I don’t let them write on the blackboard. I, I probably should; I’m —
[LAUGHS]
Strogatz: But I, I tell someone no, don’t do —
Kontorovich: You control the chalk?
Strogatz: I, no, I don’t, we don’t use chalk. I want them to write on paper, so that I can then have the paper —
Kontorovich: Ah —
Strogatz: — afterward, and look at that. But a lot of mathematicians —
Kontorovich: Yeah.
Strogatz: — insist on using chalk on blackboard, in which case, nowadays, with cellphones, you could just take a picture of their blackboard.
So why do you even take notes? Why aren’t you just taking photographs?
Kontorovich: There’s something about writing something down on a piece of paper.
Strogatz: Sure.
Kontorovich: I, I don’t know, I don’t know if it’s, it’s just at the research level — certainly for students, when I see students taking a picture of my, my blackboard, when I’m, you know, when I’m lecturing —
Strogatz: Yeah.
Kontorovich: — I immediately tell them, “You have learned nothing.”
[LAUGHS]
Kontorovich: “You will never look back at that picture, and if you do, it’ll be a bunch of jargon that you won’t understand anything that’s there.”
Strogatz: Uh-huh?
Kontorovich: “You must write it down with your own hand. The physical act of moving the pen will start to ingrain that idea into your brain.” Like, what I’m trying to do is get your neurons to flash at a certain, you know, frequency, in the certain places —
Strogatz: [LAUGHS] I think you’re right, uh, I think a lot of us feel that, right?
Kontorovich: Yeah.
Strogatz: That notetaking, as a physical act, does something different than take — much different —
Kontorovich: Yeah.
Strogatz: — than taking a photograph, or just listening.
Kontorovich: What’s the point of a college education anyway?
Strogatz: Good question.
Kontorovich: Uh, they can, right, they can google, uh, any of this stuff, they can watch YouTube videos.
Strogatz: All right, so what’s —
Kontorovich: It’s the fact that there’s a human being standing there at the blackboard, who might, at some crazy time, point a finger at them and ask them a question. And they have to react on the spot, or they have to, you know — like, that’s, I think, what they’re getting.
Strogatz: All right, but you’re tantalizing us, now, with that question of yours:
[LAUGHS]
Strogatz: What is the point of a college education? That’s something that many people think about, and especially where college is so expensive, and the job market is so tight. I mean, you — is it the knowledge?
Kontorovich: Ah —
Strogatz: Some people would say you go to college for the knowledge, some would say you go for ways of thinking. You’re going to forget all that knowledge or it’s all going to become obsolete in your lifetime. What would you say?
Kontorovich: Well, I would like to think that studying mathematics rigorously is a path to — not happy life in the sense of, you know, I don’t know if happy is a good word — fulfilled life.
Strogatz: Uh-huh?
Kontorovich: You know, it, it helps you think about complicated decisions — in your life, you will have to make a number of complicated decisions, you know, who you get married to, and what you do with your career, and if you have children — you know, there’s so many, uh, important life-altering decisions that you make.
It, it’s like, it takes stamina, it takes — uh, if you never develop that stamina, through some kind of rigorous training —
Strogatz: Mm —
Kontorovich: — then I would say you have a disadvantage.
Strogatz: Mm-hmm, that’s interesting.
Kontorovich: Um —
Strogatz: Because a lot of people would say, uh, anything about your actual life, like, who to marry or where to live or what job to take or — that those things should be done by gut reaction. That if you try to do it too logically, using your math-style training, you know, pros and cons, adding up the weights of the things you care about, that’s not really how anybody makes important decisions in their life. They, they do it by…
Kontorovich: Of course.
Strogatz: But, but your stamina argument is interesting, that having done anything hard, including trying to do math —
Kontorovich: Yes.
Strogatz: — gives you some stamina, it gives you the stick-to-itiveness, the tenacity —
Kontorovich: Yes, the mental stick-to-itiveness.
Strogatz: — that are going to help you in all parts of life?
Kontorovich: Yeah.
Strogatz: Really?
Kontorovich: Yeah.
Strogatz: That’s, so that’s an interest — I mean, I, I’m asking because I, I wrestle with this question a lot. I mean, all, anyone who teaches math [LAUGHS] has to confront the question where you know most people are not going to use the math we teach them.
Kontorovich: Right.
Strogatz: And, so, what are we doing? And all math teachers in high school, and even elementary school — I think everybody knows you need to know, when, when you go get your blue jeans at the GAP and it says 20 percent off, what does that mean, okay, you want to know what that means. But you certainly —
Kontorovich: Well, uh, maybe not even, because the, the cash register will ring it up and they’ll tell you, right? [LAUGHS]
Strogatz: That’s true, they’ll tell you. So, so there’s a real serious question, what’s the point of all this math? And especially in light of the fact that math is an obstacle for many people, and it keeps them out of community colleges, or it causes them to — you know, there are people who are very hung up on math and will say, “I don’t get any — like, why do I, as — I want to be a ballerina or I want to be, you know, in some field that doesn’t use math at all —
Kontorovich: Yeah.
Strogatz: — “why, why are you torturing me with this stuff?”
Kontorovich: Well, by the same token, why does anyone need to read, read, first of all, and read, certainly, read fiction?
Strogatz: But everyone knows why you need to read. That’s — but read fiction is a fair question.
Kontorovich: But why do you need to read fiction? Yeah.
Strogatz: Why do you need to read Shakespeare, or why do you need to read Toni Morrison —
Kontorovich: Yeah.
Strogatz: — you know, or whatever?
Kontorovich: Don’t you have a more fulfilled life — why do you need to watch Harry Potter movies?
Strogatz: Well, you don’t.
Kontorovich: Don’t they bring you — but, uh, you don’t, but if you do, then, don’t you get a certain amount of satisfaction —
Strogatz: Sure.
Kontorovich: — struggling with the story and —
Strogatz: I mean, math is important in a way that Harry Potter is not —
Kontorovich: Yeah.
Strogatz: — or in a way that chess is not. I mean, math and chess are comparably difficult —
Kontorovich: Yeah.
Strogatz: — but, but chess is, is pointless in a way that math is not. It seems to be, as someone who loves both of them.
Kontorovich: Yeah, on the other hand, if schools mandated chess, uh —
Strogatz: That’s right.
Kontorovich: — I wouldn’t be opposed.
Strogatz: I agree. And actually, when I used to play chess, um, very seriously, in tournaments and stuff, I learned something from chess that has been more valuable to me in my life than anything I learned in math. Which was —
Kontorovich: Uh-huh?
Strogatz: — you have a position on the board, and that’s the position you have.
Kontorovich: Yes.
Strogatz: And you may have screwed up to get to that position, but you’re not dead, yet.
Kontorovich: Right.
Strogatz: And you’ve gotta do the best thing you can do, in the position that you’ve got.
Kontorovich: Absolutely.
Strogatz: And, and to the extent that you keep thinking about how you got yourself messed up to get into that position, you’re not going to be thinking about what you need to dig out of the hole.
Kontorovich: Right.
Strogatz: So, just focus on what you’ve got — and you can often get lucky and make a, do a trick and the other guy will make a mistake, and so on. And so, that kind of feeling of having your wits about you even when things seem desperate —
Kontorovich: Yeah.
Strogatz: — has been very valuable in real-life, more, more —
Kontorovich: Yeah.
Strogatz: — than any theorem I know.
Kontorovich: And for, especially for younger kids, I think it’s so important. I took my, my older son to his first chess tournament, and he lost the game.
Strogatz: Yeah.
Kontorovich: And, you know, being able to deal with, with that, in the, in the days of everybody gets a participation trophy. And he got a, I mean, he got a trophy anyway, but, uh, being able to lose a game, that is such an important lesson to be learned —
Strogatz: Uh-huh.
Kontorovich: — there, that, uh, math also could teach, if you, you know, if a, if a problem beats you, if you can’t solve it.
Strogatz: I mean, you had made the point that you liked the, the feeling, this reassuring or pleasing feeling of knowing that, that something that’s true in math is true forever.
So that’s one of the things. The satisfaction of solving a hard problem, is that — that sounds like that’s part of it, for you, too.
Kontorovich: Yeah, overcoming adversity, like, our entire species is built on chasing down that deer.
Strogatz: Yeah.
Kontorovich: And if you don’t get the feeling, after chasing a deer down for five hours and finally getting it and eating it, that, like, “That was really, really worthwhile and that felt really good, even though I was running for five hours,” you know, the next time I see a deer, I might be, like, “Oh, god, I have to run after this thing for five hours?”
Strogatz: Yeah, I know, but you’re giving me this made-up evolutionary story now. I want to know your, your thing.
[LAUGHS]
Strogatz: Like, did you actually chase down a mathematical deer? Give me a real story.
[LAUGHS]
Kontorovich: Sure, yeah, problems that I’ve worked on for five years that resisted all effort, and then eventually, you know, there was some breakthrough from some other idea, yeah.
Strogatz: Did that really happen, five years?
Kontorovich: Um — yeah. I mean, imagine, like, a, a two-foot thing of stone, and you have a little chisel, and the first —
Strogatz: Oh, you’re giving me a metaphor again. I want the real story. [LAUGHS]
Kontorovich: I don’t know — the real story.
Strogatz: Yeah.
Kontorovich: Well, that kind of is what the real story feels like. There’s this big thing —
Strogatz: Yeah?
Kontorovich: — you know, it’s a 100-page paper, and how do you write a 100-page paper? Well, you write it one paragraph at a time.
Strogatz: Yeah.
Kontorovich: Um, but, I mean, there’s the paper and there’s the proof, right? The, the first proof is 300 pages, until you figure out what the hell you’re doing.
Strogatz: Wow.
Kontorovich: Uh, because there’s all the wrong turns.
Strogatz: You’re not kidding? It’s really that big?
Kontorovich: Uh, uh, if you literally take all the notes that, that —
Strogatz: Yeah.
Kontorovich: — were, were written down, and all the false, uh —
Strogatz: Sure.
Kontorovich: — false turns —
Strogatz: Wow.
Kontorovich: — um, yeah, I mean, at least — I don’t know, thousands of pages. I never tried to count them.
Strogatz: This is hard work. [LAUGHS]
Kontorovich: [LAUGHS] I mean, it better be, because [LAUGHTER] nobody figured it out, before, right?
Strogatz: I guess so.
Kontorovich: It wasn’t something where it’s just a couple of pages and you get the idea.
Strogatz: Oh, but —
Kontorovich: So, actually, from the eagle’s view, the idea is very simple, um, and it’s, and it’s the right way of — it’s like the Archimedes, how do you chop things up and weigh them.
Strogatz: Yeah.
Kontorovich: Uh, and then, when you try to make the details work out, that’s when you start running into massive amounts of trouble.
Strogatz: Uh-huh.
Kontorovich: And then, every time, you know, you’re trying to put the carpet in the room, and you stick one corner in, then another corner pops up over there somewhere.
[LAUGHS]
Kontorovich: And you can’t quite get it.
Strogatz: Yeah, the whack-a-mole method of, of solving math problems.
Strogatz: There are two styles that the absolute greatest seem to have. Some people would say that a man named Alexander Grothendieck was the greatest 20th Century mathematician. Grothendieck has this description of how to solve a hard problem. Imagine like it was a, a nut that you’re trying to crack, and it’s a really hard resistant nut. You could come out there with a chisel and a hammer, and start smashing away at the nut until you crack the shell and then get the nut. That would be like the Bourgain style. But the, the Grothendieck style is you would take the nut and put it in warm water, and leave it in there for days. And then you come back, after the shell is soft, and you just gently open the shell and there’s the nut. What that means in math terms is that Grothendieck was known for developing super advanced theories that made every problem as soft and easy to crack as this nut in the hot water.
Strogatz: Whereas, the, the Bourgain style is you don’t have patience for that, and you come at with everything you’ve got, and you smash it open.
Kontorovich: There were so many times where we hit an absolute wall —
Strogatz: Yeah?
Kontorovich: — and the resolution was, “You don’t actually need to cross this wall.”
Strogatz: Aha.
Kontorovich: “We don’t have to resolve this obstacle, to get to our end goal.”
Strogatz: Different roads —
Kontorovich: There’s, like —
Strogatz: Different roads leading to the desired place.
Kontorovich: Yes.
Strogatz: So that’s a resourcefulness that you’re talking about —
Kontorovich: Yes.
Strogatz: A feeling of don’t —
Kontorovich: He was extremely resourceful in that sense.
Strogatz: — don’t keep banging your head on that wall. You can look at another wall to bang your head on.
Kontorovich: Yeah. When we were stuck, you know, when we really were stuck, we would be at, at the blackboard for six hours, we’d be utterly destroyed, everything we tried, nothing worked, like, I didn’t — I don’t — I saw the human side of this guy.
Strogatz: Yeah?
Kontorovich: He wasn’t just this, uh, angel that somehow proved math theorems with his pinky.
[LAUGHS]
Kontorovich: Like, he worked his ass off, that’s why he —
Strogatz: Oh —
Kontorovich: — he was so successful.
Strogatz: Huh.
Kontorovich: But when, you know, we would be totally stuck, and then he would have some new idea, and the new idea was, “We’re looking in the wrong place.”
Strogatz: Beautiful.
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Next time on “The Joy of x,” neurobiologist Leslie Vosshall teaches me to respect my nose, and sends me on an unexpected mission.
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“The Joy of x” is a podcast project of Quanta Magazine. We’re produced by Story Mechanics. Our producers are Dana Bialek and Camille Petersen. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta Magazine, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet, and at the Cornell University Broadcast Studio, Glen Palmer and Bertrand Odom-Reed, though I know him as Bert. I’m Steve Strogatz. Thanks for listening.
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