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THE JOY OF WHY | VIEW ON WEBSITE

How Does Graph Theory Shape Our World?

By Janna Levin +1 authors

Steven Strogatz

June 26, 2025

Maria Chudnovsky reflects on her journey in graph theory, her groundbreaking solution to the long-standing perfect graph problem, and the unexpected ways this abstract field intersects with everyday life.

Peter Greenwood for Quanta Magazine

Introduction

Born in the 18th century when Leonhard Euler solved the puzzle of the seven bridges of Königsberg, graph theory has become a foundational tool in mathematics. It studies relationships through nodes (vertices) and the links (edges) that connect them, transforming the complexity of systems — from friendship networks to airline routes — into elegant abstractions that reveal underlying structure and interaction.

Maria Chudnovsky from Princeton University is a leading mathematician in the field. In this episode of The Joy of Why, Chudnovsky talks with co-host Janna Levin about how she got into graph theory, solved the decades-old perfect graph problem, and used it to plan her wedding seating chart. Chudnovsky also reflects on her appearance in commercials as a “superstar mathematician,” and how her background primed her for a discipline that transcends language, culture and time.

Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.

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Transcript

JANNA LEVIN: I’m Janna Levin

STEVE STROGATZ: And I’m Steve Strogatz.

LEVIN: And this is The Joy of Why, a podcast from Quanta Magazine exploring some of the biggest unanswered questions in math and science today.

LEVIN: Steve, hi.

STROGATZ: Hi there. How are you doing, Janna?

LEVIN: Good. It’s good to see you again.

STROGATZ: Yeah, great to see you.

LEVIN: I have a math problem for you.

STROGATZ: Oh, love it.

LEVIN: So, imagine you’re visiting a city, and it has a river, maybe let’s say. Maybe it’s an island like Manhattan, and there’s little islands like Randall’s Island or Roosevelt Island, and they’re connected by bridges. And you’re a tourist, you want to cross every bridge, but you don’t want to overexert yourself. So, you’re going to try to cross each bridge once.

STROGATZ: Mmmmm.

LEVIN: Have you heard of this before as a mathematical problem?

STROGATZ: You know, I have.

LEVIN: Yeah, seed planted.

STROGATZ: Yeah. Yeah. No, it’s a classic problem sometime in the 1700s. Leonhard Euler was asked this question. You know, a funny thing about it that I only recently learned is that when the mayor of that town, Königsberg, asked him to think about this question, Euler complained, what are you asking me for? This is not a math problem.

LEVIN: Right.

STROGATZ: Isn’t that interesting? To him, this was not math at the time.

LEVIN: Yeah, that’s really interesting.

STROGATZ: He founded a new branch of math in thinking about it.

LEVIN: Right

STROGATZ: But he didn’t see it as math at the time.

LEVIN: So how did he get interested enough to mathematize the problem, which is really what we’re doing a lot of the times?

STROGATZ: I think it’s just that Euler felt that it was a curious question worth thinking about, just because it involved, as he put it, reasoning.

LEVIN: Mmmmm, a puzzle.

STROGATZ: It was a puzzle, yeah.

LEVIN: Yeah, now Königsberg is now the Russian city of Kaliningrad, if I’ve got that right. So, can a visitor there figure out how to go across? There’s a river, there’s a mainland, there’s two islands and seven bridges.

STROGATZ: Yeah, that’s right. And in that old problem, the townspeople used to go out on Sundays and take strolls, and it was an amusement to see if they could walk across each bridge exactly once. And no one could ever do it, but nobody could prove it was impossible. Until Euler, who did prove it was impossible.

LEVIN: And so how does he turn this into a math problem?

STROGATZ: Well, if you look at his original paper, what he does is use letters. Capital letters for the land masses and lowercase letters for the bridges.

LEVIN: Hmmmm.

STROGATZ: And then he plays around with the combinations of these uppercase and lowercase letters that would correspond to admissible walks.

LEVIN: Oh, how interesting.

STROGATZ: Nowadays, we would use dots and lines. So, and of course he found that he couldn’t do it because some of the land masses had an odd number of bridges touching them.

LEVIN: So, once he solves that problem, he realizes, oh, immediately he can say, if this problem crops up again, you need an even number of bridges. So, he’s already…

STROGATZ: He’s already generalizing.

LEVIN: … making headway towards generalizing. And so, you said it invented a whole new branch of mathematics, and what’s that branch?

STROGATZ: Well, we call it graph theory, or if we’re in England, we would say “graRph theory.”

LEVIN: Well, here we were talking to Maria Chudnovsky, who’s a really interesting mathematician at Princeton. She works primarily on graph theory and managed to solve a very interesting, outstanding problem, it was nearly 50 years old, about perfect graphs.

STROGATZ: Hmm, that’s a term I haven’t heard before.

LEVIN: I’ll let her sort of tell the story, but it’s really about color coding and visualizing and nodes and dots and connectors. So, let’s hear from Maria. You want to eavesdrop a little bit?

STROGATZ: I’m all ears.

LEVIN: Great. Here’s Maria Chudnovsky from Princeton talking about graph theory.

Maria Chudnovsky

[Music plays]

LEVIN: Maria, welcome to the show. It’s so nice to have you on.

CHUDNOVSKY: Thank you.

LEVIN: I wish we could be in the same studio. We realized we’re only a few blocks apart.

CHUDNOVSKY: Maybe next time.

LEVIN: Maybe next time. I actually was amused to discover that you had been in a TurboTax commercial.

CHUDNOVSKY: That’s right. That’s my claim to fame. I also did a mattress commercial, did you know that?

LEVIN: Yes, I saw you sleeping in a mattress commercial.

CHUDNOVSKY: Exactly, exactly. “Smart begins with better sleep.” There’s no statement I believe in more.

LEVIN: They’re getting free advertisement with us here. I was amused to see that they listed you underneath your name in these commercials as a “superstar mathematician.” And I wonder if you think the culture is changing in our attitude towards mathematicians?

CHUDNOVSKY: I think there’s a lot more understanding now of how important math is. I used to tell people I’m a mathematician, “They would say, oh, yeah, I’m terrible in math. Math is really boring.” And now they say, “Oh, wow, that’s really interesting, what do you do?”

I think, you know, with the advance of technology and the internet and, obviously our lives have changed a lot over the last 30 years or 100 years, depends how you want to think about it. And I really think math is kind of gaining popularity and gaining traction.

LEVIN: It really is interesting that mathematical literacy or scientific literacy is considered important, more than it used to be, at least.

CHUDNOVSKY: I’m so happy about that because, you know, it really it seemed to be an educated person, you need to read books, you need to know something about art, and I would never say that’s not important. But now it seems that there’s another component to it. And, you know, given that that’s a huge part of my life, I’m very, very happy about that.

LEVIN: There’s also an old idea that mathematics is the language of the universe, which might actually be attributed to Galileo. What do you make of this quote? I mean, do you see it as a universal language of sorts?

CHUDNOVSKY: I certainly do in many ways. Like, you don’t need to speak any particular language very well in order to be able to effectively communicate mathematics. If you have something interesting and important to say in math, you need to have some basic common language as the person you’re speaking with. But you don’t need to be, um, eloquent. You don’t need to be a great speaker. You can just write down what you have to say and say what you have to say, and it’s been communicated. There is no kind of added layer of “I need to say it in such a way that it convinces them.” If you have a proof, it convinces them. It doesn’t matter how you say it.

LEVIN: And it’s the same around the world.

CHUDNOVSKY: And it’s the same around the world. Yeah, there’s really no differences. Another thing math brings to the table is clear thinking.

That’s maybe a little less of a language thing, but the way you understand things, the way you think about things, you try to break it into pieces and see what’s important, what matters, what was incidental and coincidental and not important. If you’re used to thinking in a mathematical way that really puts you at an advantage.

LEVIN: It’s like a meta-language, like a way of understanding the world that transcends the grammatical structures.

CHUDNOVSKY: Right, there are all these studies that depending on what language you think it, you perceive the world a little differently—which I’m incredibly fond of.

LEVIN: That’s amazing. I was actually going to ask you about that hypothesis in linguistics that language structures your thoughts.

CHUDNOVSKY: So, I’m fluent in two languages and I understand three languages. And I know this about myself; things change a little bit in the way I think when I’m dealing with a different language.

LEVIN: Fascinating. I understand that you did have a personal experience with languages, with countries, with moving. You were born in Russia and moved to Israel as a teenager. So, this played a personal role for you, this issue of language, transcending language and using math as a medium to express yourself.

CHUDNOVSKY: Absolutely. I’m sure the fact that of all the subjects I could study, the one that required the least command of a language was math, had a lot to do with my choice of field of study.

As a mathematician and a scientist, if you’re used to thinking mathematically, then, you perceive the world in a certain way, which is a little bit like if you’re used to thinking in a certain human language, then you perceive the world in a certain way.

LEVIN: I don’t want to completely leave this idea too quickly. I wonder, how math can structure your thoughts, the particular field that you choose can structure your thoughts. You’re in a branch of math known as graph theory. Can you give us just a quick explainer of graph theory to anchor us in your subject?

I only think in pictures.

CHUDNOVSKY: Sure. Graph theory is branch of math that studies pairwise relations. Let’s start with something in the world. Suppose you have a system that consists of objects, and some pairs of objects are in a relation, and some are not.

Like, you can have a bunch of people, and some pairs of people are friends, and some pairs of people are not friends. Or you can start with, buildings and roads between them. Or you can have a bunch of cities, and some pairs of cities have direct flights and some pairs of cities don’t have direct flights between them. So, all these things are examples of graphs where you can take objects and then say this pair is in relation, it goes together, this pair doesn’t go together.

So now, forget all these examples and just stay with the abstraction where you put a dot on your piece of paper for every object and now you put a line between two dots — we call it an edge — if these two objects were in a relation.

So now you have an abstract mathematical object, called a graph. Apparently, Euler came up with this idea in 17-something. And now instead of studying a system of people’s friendships or a system of flights or a system of roads, you can just study this abstraction of a graph. You’re no longer a social scientist or a transportation expert. Now suddenly you’re a graph theorist.

LEVIN: Right, you’re a mathematician. What drew you to this particular topic of graph theory? And also, do you really think you think more visually and differently because you’re so embedded in graph theory, or you think it was the other way around? You were inclined to visual thinking and that drew you to graph theory?

CHUDNOVSKY: I think the answer is we’ll never know. When I was applying for grad school, I knew I wanted to do discrete math because when I was in college, these were the courses that somehow were easiest to me. I felt that I understood them the best. Like when you take a class, some classes you study, you prepare for the test, you get your grade, and you’re grateful it’s done.

Some classes, they really stay with you, and you keep thinking about it. It’s kind of just the tip of the iceberg, and then you keep going deeper and deeper in your mind, and then when you show up for the test, you didn’t even need to study, because you understood it so much better than what they tried to teach you. And so, classes in discrete math were like that for me.

LEVIN: I personally love discrete math. I once had a friend who said, “there are different kinds of math?” Like this is a surprise to people who don’t live in this world that there are different kinds of math. Can you help our audience with the difference between, sort-of, continuous and discrete?

CHUDNOVSKY: Sure, right. Roughly, you can partition all of mathematics into two parts. One part is things where you can say, “this one is the next one after that one.” And then things where you can’t say that, because between every two things, there’s another one. So, I’ll give you an example, though, because of course if you didn’t know what I meant before, you don’t understand now either.

Let’s say I’m thinking about, counting numbers, natural numbers. So immediately after one comes two, immediately after two comes three. and that’s discrete. There’s no counting number between seven and eight.

Now let’s think about fractions. Between one and two, there’s one and a half. Between one and one and a half, there’s one and a quarter. Between one and one and a quarter, there’s one and an eighth. I’m not going to continue, but I think we get the gist.

The first example was a discrete example. The second example is a continuous example.

Courses in discrete math were somehow better for me. I always enjoyed them better; I understood them better. And so, when I was applying to grad school, it was clear to me that I was going to do some kind of discrete math. Then I got into Princeton, where the kind of discrete math they were doing was graph theory. And I became a graph theorist. Had I gone to a different university, I would have probably done something else, and maybe I would have been good at that, or maybe I would have had a different career.

LEVIN: You might not have made it into a mattress commercial.

CHUDNOVSKY: Exactly.

LEVIN: One of the aspects you mentioned, just as an aside, I think is so fascinating about the continuum between two integers is you can remove an infinite number of rationals and still have an infinite set left over. I mean, the beauty of discrete math is really mind-blowing.

On another note, I think it’s interesting for people that graph theory is a very visual subject. I often see that you’re drawing pictures on the board and making connections. You’re doing them very fast because you’re so used to it. But how strong is the visual aspect for you in your actual work? How much are you actually using a kind of visual mapping do you think?

CHUDNOVSKY: All the time. I only think in pictures. Eventually, I have to sit down and write it because, otherwise you miss things and you make mistakes, but all the development of ideas is visual.

LEVIN: And it’s quite beautiful to watch. Now, a lot of people also wonder what mathematicians are doing all the time.

CHUDNOVSKY: Doodling.

LEVIN: You know, what’s at stake for the mathematician? Why are these problems important? There’s this sort of tension between the very abstract mathematics, which I believe you very much love, and real-world application, but what is that balance like for you? What’s really at stake for you as a mathematician?

Math is math and truth is truth.

CHUDNOVSKY: I mean, I do abstract math. I’d be very happy if somebody someday finds applications for what I do. You know, sometimes if people come to me and they have a question I can help them with, and it has something to do with the world, that’s exciting. That’s fun. I like it. But my world is completely abstract.

LEVIN: Do you ever have to use graph theory to solve your own real-world problems?

CHUDNOVSKY: So, when I was getting married to my husband, we had to design a seating chart. And we were nervous about it. And I said to him, let me try to do it. You can’t be too picky about it. But I took our list of guests, which, by the way, the vast majority of guests at our wedding were people I didn’t really know from my husband’s side, I only knew kind of some stories about them.

But it’s a graph theory problem. What you do is, you make a graph where the vertices, the dots are the people. And now you put an edge between two vertices, if these two people are such enemies that you can’t put them at the same table.

And now what you need to do is take your graph and partition its vertices into subsets so that in each subset, you don’t have two people assigned to the same table that can’t sit together. You don’t have two vertices joined by an edge. And, if there are a lot of edges in your graph, that’s a hard problem. But if there are few edges in your graph, namely most of your friends and family get along, then it’s actually easy. You can use something called the ‘greedy algorithm’ and it just works.

You know, our friends and family are reasonable people. There are not so many edges in this graph. So, I took this list and a few minutes later I had—it’s called the coloring of the graph—a seating arrangement, tables of 10 to 12. And it was basically completely acceptable. My husband was very surprised by that.

LEVIN: Your wedding went without incident, I assume?

CHUDNOVSKY: No. No, no incidents.

LEVIN: Now, you’ve just been describing graph coloring, right. You’re color coding. This is the group and here’s a different group. As soon as you get to a point where, “Oh, this person can’t sit next to this person,” you need to assign a new color. Now this, graph coloring, is this a very old human problem?

CHUDNOVSKY: It absolutely is. There is probably the most famous problem in graph theory, the Four Color Theorem, which was actually posed by a cartographer 200 years ago. And what he posed is that for any map on earth, you can color in such a way that two countries that share a border get different colors, right? So, you can do that by using at most four colors. No matter how the world is partitioned into countries you can color it with four colors in such a way that countries that share borders get different colors.

LEVIN: It’s not obvious.

CHUDNOVSKY: It is very, very far from obvious. It inspired a lot of mathematical research for dozens of years. It’s very far from being trivial.

But let me just say a few things. One is when you say two countries share a border, they really need to share, a little interval of a border. Sharing just one point doesn’t count.

Thing two, this is only true if the countries are connected. If you have country with islands, that’s false.

And the third thing is, you know, I said every map on Earth, I amuse myself with that. That’s not just an expression. If your map is not on Earth, if your map is on a planet that has high genus, like a donut or a pretzel, then it’s not true.

LEVIN: Mmmmm. So, if it’s multiply connected in some way.

CHUDNOVSKY: Right, not simply connected.

LEVIN: So how fascinating. So, there is right there a real-world application, which is trying to understand maps and borders. Maps in the 19th century were very important in terms of beginning to understand and think spatially. You’ve had a very big impact in an area that describes perfect graphs. Can you explain to me what makes a graph perfect in terms of coloring?

CHUDNOVSKY: I’m so glad we started this by talking about graph coloring, because this sets the stage perfectly for talking about perfect graphs. So, you have a graph. It has vertices and edges. And now you would like to color it. So, you want to partition the vertices into sets so that there’s no edge inside a set, right? Another way to say it is you color the vertices with different colors. And the rule is vertices of the same color are not joined by an edge. If two vertices are joined by an edge, they get different colors.

Okay, so far, so good. But here is a coloring. Color every vertex with a different color. That is not a very interesting concept. To make it a little less abstract, imagine you know, you have to pay per color and you don’t want to buy more paints than you need. Okay, so now the question becomes: If I give you a graph, how many colors do you need in order to be able to color it? That’s called the chromatic number of a graph.

First of all, algorithmically it’s very hard. If I give you a graph and ask you how many colors do you need? What’s the smallest number of colors with which you can color this graph? You in general can’t give an answer in a reasonable amount of time.

Okay, but another thing is maybe I can look at a graph and start connecting how many colors I need to add the properties of this graph. Maybe I still won’t be able to give you an answer, but here’s an obvious lower bound on the number of colors you need. If I have a graph and in it there is a hundred vertices all pairwise adjacent, I need at least a hundred colors in order to color this graph. Now, there are graphs with no three pairwise adjacent vertices that need a thousand colors, and there are graphs with no three pairwise adjacent vertices that need a million colors.

Now, on the other hand, there are graphs where that doesn’t happen. There are graphs where this obvious lower bound of how many pairwise adjacent vertices you have is actually the right answer. And these are called perfect graphs.

LEVIN: I get the gist of it and let me just try to help for our audience to bridge a little gap. It’s my understanding that when you have these, you’re calling them pairwise adjacent vertices, we could call those a clique.

CHUDNOVSKY: Yes. Clique is the right word for it.

If you’re used to thinking in a mathematical way, that really puts you at an advantage.

LEVIN: I’m just imagining that socially people understand when you’re trying to do a wedding graph or a dinner party graph that there’s a little clique. And in that clique if that number of vertices equals the number of colors you need to use, the chromatic number, then you’re on your way to a perfect graph.

CHUDNOVSKY: So, a graph is perfect if its chromatic number equals to the largest size of a clique in it.

LEVIN: Oh, excellent. Exactly. So, the largest clique, yes. Now, in the 1960s, a French mathematician, Claude Berge, began to think more about coloring and perfect graphs. He formulated this conjecture known as a strong perfect graph conjecture. Now this has loomed large in your work. How does your work tie in with this famous conjecture, long-lasting conjecture, over 50 years?

CHUDNOVSKY: So, when I came to Princeton as a graduate student I came to the office of Paul Seymour. And he was working on the strong perfect graph conjecture at the time, and I knocked on his door, and I said, “Can I work on this with you?”

And I think he was a little surprised, because that’s not a question people often ask, but he didn’t know what to say, so he said yes. And then a couple of years later, four of us solved this problem. It wasn’t my first theorem, but it was the first theorem I proved as a PhD student and, you know, it was a good way to start a career.

LEVIN: That must have had an enormous impact — a half-century-old conjecture, many people working on this important problem. How did you approach this, you and your collaborators, so that you were able to make progress where others were stymied?

CHUDNOVSKY: That problem was open for 40 years at the time, and people worked on it. Berge’s conjecture was every graph with this and that structural property is perfect. And it was clear that, if and only if, a graph is perfect, then it must have those structural properties. You know, when you just say a sentence like that, you have no idea where to start, right? How would you prove that everything with those three properties have that fourth property?

But what you could try to do is try and understand what all graphs with these first three properties look like. And then take that and deduce from that the fourth property you’re actually interested in.

And because people had been thinking about this problem for 40 years, there were a lot of ideas of how this gap may be bridged. None of them were exactly correct. In the end, we had to come up with our own correct intermediate statement. But there was enough thoughts in the world that at least we knew in which direction we should start going. And then, you know, we worked really hard, we got lucky, we are smart, all kinds of things.

LEVIN: So, I love how that refers to a larger global collaboration, which I think relates to something we were talking about in the beginning about the transcendence of mathematical language, that this is an international collaboration, even if not in an individual paper, in the larger scheme of things.

CHUDNOVSKY: Very much so. I mean, I work with people from all over the world. I travel a lot. And because math is math and, truth is truth you can disagree on everything, but you will agree that the proof is correct, or you will agree that the proof is not correct, and your other opinions don’t play into it at all.

[Music plays]

STROGATZ: I love how she says it. Math is math and truth is truth. But it makes me wonder if this kind of training that we get prepares us ill for the real world.

LEVIN: Maybe you could say the real world is overrated.

STROGATZ: You know, that’s what some of us think in math because how many other places can you say something like that? Math is math and truth is truth. I mean, this is an oasis. It’s such a pretty place to live.

LEVIN: It’s such a pretty place to live. And we talk about this a lot on the show; just how transcendent the ideas are. They’re true for all of us, and that’s really special and really unusual.

STROGATZ: It’s also a gift that we get to talk to each other across the centuries. You know, like we mentioned Leonard Euler, and you can have in your mind, at least a conversation with someone from hundreds of years ago. There are some differences, but there’s a lot that we have in common.

LEVIN: Absolutely. I love that idea that we’re having a dialogue across time. I mean, not only across cultures, right?

STROGATZ: Right.

LEVIN: Not only across the globe, but we can actually go across time. I mean, this is, when we talk about sending messages out into the universe the most reasonable messages to send are ones that have to do with math.

Let’s go back to our guest. Let’s hear more from Maria right after this break.

[Music plays]

LEVIN:  Welcome back to The Joy of Why. We’ve been speaking with graph theorist Maria Chudnovsky on how she proved the strong, perfect graph theorem, something that was proposed by Claude Berge back in the 1960s.

I do abstract math. I’d be very happy if somebody someday finds applications for what I do.

Math can be very frustrating, and it can take a very long time. You have this enormous breakthrough, but it’s not a single moment. For you, how does this unfold, and come clear to you that you were really onto something?

CHUDNOVSKY: You know, you never know until you know. You really think you’re making progress, you set your next goal and you proved it, and now you think, “Okay, well, great, this was a big hurdle, now it’s a straight line from here,” and then suddenly it’s not a straight line. It’s a nerve-wracking process.

But on the plus side, there are a lot of little celebrations. Just because it’s not the last hurdle doesn’t mean it wasn’t an important one, right? Every step is a eureka moment. I think now every proof is very complex. There’s not one eureka moment. But there are a lot of victories. There are a lot of celebrations. Ingrid Daubechies in some interview once said, “the reason we do math is because we’re addicted to the high.” And it’s completely true.

LEVIN: That’s a t-shirt. I don’t know if this story is true or if it’s apocryphal, but I heard that Berge was read the news of the impending proof while he was in the hospital. Is that a true story?

CHUDNOVSKY: You know, he was in the hospital, that’s true. He was told the news in his last days when we found the proof. He was told about the proof.

LEVIN: It’s a nice dimension, and of course, mathematics is ultimately a human pursuit. I wanted to ask a little bit about applications that have branched off of this discovery. Are there, to your knowledge, examples in which people are using this in other fields?

CHUDNOVSKY: Right, so the answer is I don’t know. There is a paper about perfect garbage collection in Brooklyn.

LEVIN: How grateful I shall be to you if that problem is solved.

CHUDNOVSKY: There we go. Exactly. But it’s not me, it’s before my time.

LEVIN: I definitely do not need for there to be a practical application to value mathematics. And I love that we’re asking people to appreciate that the stakes are high, even for very abstract mathematics,

I know that we have sirens in the back, so there’s probably some graph theory problem for solving the route that an ambulance or cop car has to take through Manhattan.

CHUDNOVSKY: You know, designing optimal routes is definitely a graph theory problem, right? You have junctions and roads between them, and that’s a graph, and now you’re to find best way to navigate that graph.

LEVIN: So, let’s take this back to the way you approached your solution. The problem that you approached… how long, for instance, is the paper that you wrote?

CHUDNOVSKY: It’s about 150 pages long.

LEVIN: Ok, so this is very long.

CHUDNOVSKY: This is very long. This is hard. This has nothing to do with complexity hierarchy. This is just hard.

LEVIN: This is just colloquially hard.

CHUDNOVSKY: Right.

LEVIN: How did you simplify the problem? There seemed to have been a strategy where you broke the graphs into pieces in a systematic way to allow you access to this proof.

CHUDNOVSKY: Right, so, there is a lot of breakage going on. So, the goal was to prove if you have two properties then you’re perfect. So, what we did was we showed that if you have these two properties, then either you have some graph we can describe — or what’s called, an explicit construction — or you can break your graph apart in such a way that showing that this half is perfect and showing that this half is perfect is enough.

Ingrid Daubechies in some interview once said, “the reason we do math is because we’re addicted to the high.” And it’s completely true.

So that was the theorem we proved. This is a theorem with a lot of outcomes. We have five explicit constructions, three kinds of decompositions, that’s three kinds of ways to break a graph. So, what you’re proving is if properties one and two hold, then one of those eight things happen.

Now, in general, that’s a theorem that’s very hard to prove because you don’t know which way to push, right? You know, are you trying to prove that the graph behaves like this, or are you trying to prove that it behaves like that? So, what’s helpful is to break the world into pieces, where in this part of the world it’s always like this, and in this part of the world it’s always like that. And in a way, that’s the key to almost every mathematical proof. You have some huge overwhelming thing, but you’ve simplified it into a scenario where it’s always like this, and a scenario where it’s always like this, and a scenario where it’s always like that. And if you can distinguish between your scenarios, then usually that leads you to a proof.

And so that was kind of a very important insight, a very important thing to do in this proof to understand how to approach the world in this stratified way.

LEVIN: Hmm, it’s a strategy.

CHUDNOVSKY: Right, right.

LEVIN: So, it’s fascinating. Maria, I wonder how do you know when you’re approaching a problem, how hard it is?  Is the next problem you brave going to be a necessarily even harder one, or are the challenges unpredictable?

CHUDNOVSKY: I don’t think problems are linearly ordered by difficulty. I think, every problem is hard until you solve it, and then you have a solution. The reason to choose a problem is not because it’s harder than the previous problem you solved. The reason to choose a problem is because there’s a phenomenon you want to understand, and in order to understand it you need to answer this question. If the question appealing to me, it’s an aesthetical choice. You know, it’s a little bit of a practical choice if I think I have a chance of contributing. But it’s also, like, does this question interest me?

LEVIN: And so, what is that intuition? What drives your choices in big problems?

CHUDNOVSKY: I think, and this is speculation, questions that interest us are related to questions we’ve already thought about. It’s very rare that somebody, you know, comes into the room and says, “Here is a question you must be really interested in.” If I’ve never thought about it, what does it have to do with me?

So, I think there’s like some kind of world you’re building throughout your lifetime, and every next question interests you because it relates to the world you’ve built so far. But I think it’s probably only clear to you. From the outside, I think it’s a little less obvious.

LEVIN: I wanted to ask you one last question. And that is, what brings you joy in your research?

CHUDNOVSKY: Seeing order in a new place, understanding that something is actually built in a way I can describe, even though, until recently, it was just a mess. I guess, bringing order to the mess.

LEVIN: Beautiful. Maria, thank you so much for joining us. There’s such a depth here, I could talk to you for hours.

CHUDNOVSKY: Thank you very much for having me. I really enjoyed this.

[Music plays]

STROGATZ: I’m struck by how personal the experience of being a mathematician is, because when she says that what interests her is not necessarily how difficult a problem is, but whether it connects to things that she’s thought about before, I feel like that’s not a universal phenomenon. That’s something that some of us would use.

But I’m thinking, let’s say of John Nash, the great mathematician profiled in A Beautiful Mind, who would change fields just looking for the hardest, unsolved problem. It was a macho thing for him. I think he was interested in everything as long as it was hard.

The reason to choose a problem is not because it’s harder than the previous problem you solved. The reason to choose a problem is because there’s a phenomenon you want to understand.

LEVIN: I mean, isn’t that also the testament that it takes all kinds of personalities and ways of thinking to make an entire discipline, right? A healthy discipline.

STROGATZ: Exactly. And as Maria says, it’s an aesthetic choice for her. Some people are driven by aesthetics, but I think some are driven by, I want to use my craft. You know, I have this powerful technique and I’m just looking for something I can use it on.

LEVIN: Yeah. Do you feel that challenge and difficulty intrigues you when trying to solve something?

STROGATZ: No.

LEVIN: Interesting.

STROGATZ: No, I really resonated with what she said about phenomena. It really hit me that she uses the word that a scientist would use, not a mathematician. That she sees phenomena and it’s phenomena that appeal to her, and she wants to think about things that seem curious and baffling, but that are close enough that she thinks she could contribute. So, I love that view of it. That’s much closer to my own.

I mean, actually what personally drives me has always been paradox.

LEVIN: Oh intriguing.

STROGATZ: I really like it when something seems wrong, when something is off. Like that shouldn’t be the way this works. That’s a kind of question I always like.

LEVIN: Right, and then you can’t, you can’t not follow it.

STROGATZ: Yeah.

LEVIN: You know, I think, sometimes I like these kind of big visual problems with a lot of math, right. A lot of geometry, a lot of big spatial moves. Big brush strokes that move things around.

STROGATZ: I’m taken with your analogy to painting. Do you actually paint?

LEVIN: Oh, gosh. I mean, a long time ago, but I don’t advertise myself as a painter. But it’s true. There’s a method, right? You go up and you do big brush strokes. And then you, you do, you get a refiner eventually. You’re on your two-hair brush.

STROGATZ: Well, it’s really great talking to you about all this, Janna.

LEVIN: Yeah, great talking to you, Steve. Till next time.

STROGATZ: Okay, bye bye.

Related:

1. A Very Big Small Leap Forward in Graph Theory

2. The Colorful Problem That Has Long Frustrated Mathematicians

3. Math’s ‘Bunkbed Conjecture’ Has Been Debunked

4. The Fastest Way Yet to Color Graphs

[Music plays]

STROGATZ: If you’re enjoying The Joy of Why and you’re not already subscribed, hit the subscribe or follow button where you’re listening. You can also leave a review for the show. It helps people find this podcast. Find articles, newsletters, videos and more at quantamagazine.org.

LEVIN: The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine.

The Joy of Why is produced by PRX Productions. The production team is Caitlin Faulds, Livia Brock, Genevieve Sponsler and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzalez. Edwin Ochoa is our project manager.

From Quanta Magazine, Simon Frantz and Samir Patel provide editorial guidance with support from Matt Carlstrom, Samuel Velasco, Simone Barr and Michael Kanyongolo. Samir Patel is Quanta’s editor in chief.

Our theme music is from APM Music. The episode art is by Peter Greenwood, and our logo is by Jaki King and Kristina Armitage. Special thanks to the Columbia Journalism School and the Cornell Broadcast Studios. I’m your host, Janna Levin. If you have any questions or comments for us, please email us at [email protected]. Thanks for listening.

2025 Jun 25

- A New Pyramid-Like Shape Always Lands the Same Side Up

GEOMETRY | VIEW ON WEBSITE

A New Pyramid-Like Shape Always Lands the Same Side Up

By Elise Cutts

June 25, 2025

A tetrahedron is the simplest Platonic solid. Mathematicians have now made one that’s stable only on one side, confirming a decades-old conjecture.

This shape can only rest on one of its four sides.

Gábor Domokos

Introduction

In 360 BCE, Plato envisioned the cosmos as an arrangement of five geometric shapes: flat-sided solids called polyhedra. These immediately became important objects of mathematical study. So it might be surprising that, millennia later, mysteries still surround even the simplest shape in Plato’s polyhedral universe: the tetrahedron, which has just four triangular faces.

One major open problem, for instance, asks how densely you can pack “regular” tetrahedra, which have identical faces. Another asks which kinds of tetrahedra can be sliced into pieces that can then be reassembled to form a cube.

The great mathematician John Conway was interested not only in how tetrahedra can be arranged or rearranged, but also in how they balance. In 1966, he and the mathematician Richard Guy asked whether it was possible to construct a tetrahedron made of a uniform material — with its weight evenly distributed — that can only sit on one of its faces. If you were to place such a “monostable” shape on any of its other faces, it would always flip to its stable side.

A few years later, the duo answered their own question, showing that this uniform monostable tetrahedron wasn’t possible. But what if you were allowed to distribute its weight unevenly?

At first, it might seem obvious that this should work. “After all, this is how roly-poly toys work: Just put a heavy weight in the bottom,” said Dávid Papp of North Carolina State University. But “this only works with shapes that are smooth or round or both.” When it comes to polyhedra, with their sharp edges and flat faces, it’s not clear how to design something that will always flip to the same side.

Gábor Domokos discovers and builds new shapes to understand the world around us.

Ákos Stiller

Conway, for his part, thought that such tetrahedra should exist, as some mathematicians recall him saying. But he ended up focusing on the balancing acts of higher-dimensional, uniformly weighted tetrahedra. If he ever wrote up a proof of his off-the-cuff 3D conjecture, he never published it.

And so for decades, mathematicians didn’t really think about the problem. Then along came Gábor Domokos, a mathematician at the Budapest University of Technology and Economics who had long been preoccupied with balancing problems. In 2006, he and one of his colleagues discovered a shape called the gömböc, which has the unusual property of being “mono-monostatic” — it balances on just two points (one stable, the other unstable, like the side of a coin), and no others. Try to balance it anywhere else, and it will roll over to stand on its stable point.

But like a roly-poly, the gömböc is round in places. Domokos wanted to know if a pointy polyhedron could have a similar property. And so Conway’s conjecture intrigued him. “How was it possible that there was an utterly simple statement about an utterly simple object, and yet the answer was far from immediate?” he said. “I knew that this was very likely a treasure.”

In 2023, Domokos — along with his graduate students Gergő Almádi and Krisztina Regős, and Robert Dawson of Saint Mary’s University in Canada — proved that it is indeed possible to distribute a tetrahedron’s weight so that it will sit on just one face. At least in theory.

But Almádi, Dawson and Domokos wanted to build the thing, a task that turned out to be far more challenging than they expected. Now, in a preprint posted online yesterday, they have presented the first working physical model of the shape. The tetrahedron, which weighs 120 grams and measures 50 centimeters along its longest side, is made of lightweight carbon fiber and dense tungsten carbide. To work, it had to be engineered to a level of precision within one-tenth of a gram and one-tenth of a millimeter. But the final construction always flip-flops onto one face, exactly as it should.

Explore this 3D structure with your mouse/finger.

This tetrahedron, which is mostly hollow and has a carefully calibrated center of mass, can only rest on one of its faces — a property that is difficult to achieve in shapes with straight edges and flat faces.

Mark Belan/Quanta Magazine

Explore this 3D structure with your mouse/finger.

This tetrahedron, which is mostly hollow and has a carefully calibrated center of mass, can only rest on one of its faces — a property that is difficult to achieve in shapes with straight edges and flat faces.

Mark Belan/Quanta Magazine

The work demonstrates the important role of experimentation and play in research mathematics. It also has potential practical applications, such as in the design of self-righting spacecraft.

“I didn’t expect more work to come out on tetrahedra,” Papp said. And yet, he added, the team’s research allows mathematicians to “really appreciate how much we didn’t know and how thorough our understanding is now.”

Tipping Point

In 2022, Almádi, then an undergraduate aspiring to become an architect, enrolled in Domokos’ mechanics course. He didn’t say much, but Domokos saw in him a hard worker who was constantly in deep thought. At the end of the semester, Domokos asked him to concoct a simple algorithm to explore how tetrahedra balance.

When Conway originally posed his problem, his only option would have been to use pencil and paper to prove, through abstract mathematical reasoning, that monostable tetrahedra exist. It would have been almost prohibitively difficult to pinpoint a concrete example. But Almádi, working decades later, had computers. He could do a brute-force search through a huge number of possible shapes. Eventually, Almádi’s program found the coordinates for the four vertices of a tetrahedron that, when assigned certain weight distributions, could be made monostable. Conway was right.

Krisztina Regős (left) and Robert Dawson helped discover new properties of tetrahedra.

Courtesy of Krisztina Regős; Ms. Tara Inman

Almádi found one monostable tetrahedron, but presumably there were others. What properties did they share?

While that might seem like a simple question, “a statement like ‘A tetrahedron is monostable’ cannot be easily described with a simple formula or a small set of equations,” Papp said.

The team realized that in any monostable tetrahedron, three consecutive edges (where pairs of faces meet) would need to form obtuse angles — ones that measure over 90 degrees. That would ensure that one face would hang over another, allowing it to tip over.

The mathematicians then showed that any tetrahedron with this feature can be made monostable if its center of mass is positioned within one of four “loading zones” — much smaller tetrahedral regions within the original shape. So long as the center of mass falls inside a loading zone, the tetrahedron will balance on only one face.

The gömböc, discovered in 2006, can stand on only two points, one stable, the other unstable. Mathematicians have continued to search for other shapes with intriguing balancing properties.

Gábor Domokos

Achieving the right balance between the weight of the loading zone and the weight of the rest of the tetrahedron is easy in the abstract realm of mathematics — you can define the weight distribution without a care for whether it’s physically possible. You might, for instance, let parts of the shape weigh nothing at all, while concentrating a large amount of mass in other parts.

But that wasn’t entirely satisfying to the mathematicians. Almádi, Dawson and Domokos wanted to hold the shape in their hands. Was it possible to make a monostable tetrahedron in the real world, with real materials?

Getting Real

The team returned to their computer search. They considered the various ways in which monostable tetrahedra might tip onto their stable face. For instance, one kind of tetrahedron might follow a very simple path: Face A tips to Face B, which tips to Face C, which tips to Face D. But in a different tetrahedron, Face A might tip to Face B, and both Face B and Face D will tip to Face C.

The loading zones for these different tetrahedra look very different. The team calculated that to get one of these “falling patterns” to work, they would need to construct part of the shape out of a material about 1.5 times as dense as the sun’s core.

While studying to be an architect, Gergő Almádi was drawn to a decades-old geometry problem.

Réka Dolina

They focused on a more feasible falling pattern. Even so, part of their tetrahedron would have to be about 5,000 times as dense as the rest of it. And the materials had to be stiff — light, flimsy materials that could bend would ruin the project, since it’s easy to make a round or smooth shape (like the roly-poly) monostable.

In the end, they designed a tetrahedron that was mostly hollow. It consisted of a lightweight carbon fiber frame and one small portion constructed out of tungsten carbide, which is denser than lead. For the lighter portions to have as little weight as possible, even the carbon fiber frames had to be hollow.

With this blueprint in hand, Domokos got in touch with a precision engineering company in Hungary to help build the tetrahedron. They had to be incredibly accurate in their measurements, even when it came to the weight of the tiny amounts of glue used to connect each of the shape’s faces. Several frustrating months and several thousand euros later, the team had a lovely model that didn’t work at all. Then Domokos and the chief engineer of the model spotted a glob of stray glue clinging to one of its vertices. They asked a technician to remove it. About 20 minutes later, the glue was gone and Almádi received a text from Domokos.

“It works,” the message read. Almádi, who was on a walk, started jumping up and down in the street. “Seeing the lines on the computer is very far from reality,” he said. “That we designed it, and it works, it’s kind of fantastic.”

“I wanted to be an architect,” he added. “So this is still very strange for me — how did I end up here?”

In the end, the work on monostable tetrahedra didn’t involve any particularly sophisticated math, according to Richard Schwartz of Brown University. But, he said, it’s important to ask this kind of question in the first place. It’s the kind of problem that’s often easiest to overlook. “It’s a surprising thing, a leap, to conjecture that these things would exist,” Schwartz said.

Related:

1. The Simple Geometry That Predicts Molecular Mosaics

2. Scientists Uncover the Universal Geometry of Geology

3. Why Triangles Are Easy and Tetrahedra Are Hard

At the moment, it’s not clear what new theoretical insights the model of the monostable tetrahedron will provide — but experimenting with it might help mathematicians uncover other intriguing questions to ask about polyhedra. In the meantime, Domokos and Almádi are working to apply what they learned from their construction to help engineers design lunar landers that can turn themselves right side up after falling over.

In any case, sometimes you just need to see something to believe it, Schwartz said. “Even for theoretical math, geometry especially, people are kind of right to be skeptical because it’s quite hard to reason spatially. And you can make mistakes, people do.”

“Conway didn’t say anything about it, he just suggested it — never proved it, never proved it wrong, nothing. And now here we are, I don’t know, 60 years later,” Almádi said. “If he were still alive, we could put this on his desk and show him: You were right.”

2025 Jun 23

- Matter vs. Force: Why There Are Exactly Two Types of Particles

QUANTUM PHYSICS | VIEW ON WEBSITE

Matter vs. Force: Why There Are Exactly Two Types of Particles

By Matt von Hippel

June 23, 2025

Every elementary particle falls into one of two categories. Collectivist bosons account for the forces that move us while individualist fermions keep our atoms from collapsing.

Fermions must keep to themselves, while bosons can play together in the same place.

Irene Pérez for Quanta Magazine

Introduction

Beneath the richness of our world lies a pristine simplicity. Everything is made of a set of just 17 fundamental particles, and those particles, though they may differ by mass or charge, come in just two basic types. Each is either a “boson” or a “fermion.”

The physicist Paul Dirac coined both terms in a speech in 1945, naming the two particle kingdoms after physicists who helped elucidate their properties: Satyendra Nath Bose and Enrico Fermi.

In 1924, Bose was working at the University of Dhaka, in what is known today as Bangladesh. Earlier, around 1900, Max Planck had proposed a law for how much light of each color a hot object emits. (Planck’s insight that this light comes in discrete packets, or “quanta,” set physicists on the path to quantum mechanics.) Bose found a stronger mathematical derivation of Planck’s law. He wrote to Albert Einstein, asking for help in submitting the result to a German journal, then collaborated with Einstein to flesh out the idea.

Bose and Einstein’s math described a situation where multiple particles can be perfectly alike: not just have the same charge, mass and energy but even exist in the same place at the same time. Photons, the particles of light, behave this way. A laser, for instance, consists of many photons synchronized at the same wavelength, together in a single beam. We now call such particles bosons.

The same math would turn out to work for more than just photons. Anything we experience as a force is a collective effort of uncountably many bosons. Photons combine to exert the electromagnetic force, while other bosons give rise to the forces that bind the nucleus together and cause radioactive decay. Physicists expect the hypothetical “gravitons” that produce gravity to be bosons as well. And beyond the fundamental forces, certain composite particles — for example, helium atoms — also behave like bosons.

But Bose and Einstein’s math didn’t work for the electron.

Mark Belan/Quanta Magazine

When physicists tried to analyze electrons in metal, they found strange contradictions. For example, there appeared to be an inconsistency between the way electrons carried electric currents and the way they held heat. Working independently in 1926, Fermi and Dirac both figured out what was going wrong: Electrons are not bosons. Unlike photons, identical electrons cannot pile up in the same place. Instead, each electron must differ from its comrades in at least one way: a different location, energy or orientation. We now call such particles fermions. (Another physicist, Pascual Jordan, hit on the same idea a year earlier but didn’t publish in time to share the credit.)

Fermions make the complexity of matter possible. No two electrons can occupy the same place in an atom, so the more electrons an atom has, the more they spread out into distinct layers, giving rise to the different chemical properties of hydrogen, helium, gold, silver and all the other elements of the periodic table.

Beyond electrons, the quarks that make up protons and neutrons in atomic nuclei are also fermions. So are neutrinos. And fermions need not be fundamental particles; in materials, there are groups of electrons that collectively obey the same exclusionary math, like the configurations known as Majorana fermions that might someday power quantum computers.

Satyendra Nath Bose (left) was a little-known physicist at the University of Dhaka when he devised a theory describing collectivist particles that are now named for him — bosons. Enrico Fermi (right) later developed a theory of particles that always maintain their independence, now called fermions.

Public Domain

The difference between how fermions and bosons behave in groups goes hand in hand with a second distinction between them: their spin, a measure of how they change when rotated. Bosons have whole-number amounts of spin. (Photons have one unit, for instance, and gravitons would have two.) That means that when you turn a boson in a full circle, you’ll find the same particle you started with, with the same mathematical characteristics. Fermions, meanwhile, have half-integer amounts of spin — for example, ½ for electrons. This means that after making one full rotation, an electron does not stay the same. Its mathematical representation acquires a minus sign, and you must turn it around a second time to get it back to the way it was.

These two defining characteristics initially seemed unrelated. But in 1939, Markus Fierz proved that both are consequences of the mathematical structure of quantum theory, a connection now known as the spin-statistics theorem. (His adviser, Wolfgang Pauli, published a spruced-up version of the proof the following year.)

Related:

1. ‘Paraparticles’ Would Be a Third Kingdom of Quantum Particle

2. Inside the Knotty World of ‘Anyon’ Particles

3. Physicists Create Elusive Particles That Remember Their Pasts

The proof is quite abstract, even for physicists, and it is famously hard to explain intuitively. But the upshot is that if you try to write down equations for a spin-½ particle that follows Bose and Einstein’s math, or a spin-1 particle that obeys Fermi-Dirac statistics, these theorized particles will violate sacred physical principles like causality.

The number of particle kingdoms depends on the number of dimensions. The spin-statistics theorem proves that bosons and fermions are the only two possibilities in our three-dimensional world (unless you rethink what makes two particles identical). This has to do with the fact that in 3D, a particle can turn in a spiral, passing under its old path. Spirals aren’t possible on a 2D surface, where there isn’t a notion of “under.” As a result, new types of particles called anyons can exist in 2D, with behavior falling somewhere between that of bosons and fermions. And in one dimension, the distinction breaks down altogether. In such a world on a wire, bosons and fermions are like two different equations with the same solution: the two kingdoms are secretly one.

2025 Jun 20

- Is Mathematics Mostly Chaos or Mostly Order?

INFINITY | VIEW ON WEBSITE

Is Mathematics Mostly Chaos or Mostly Order?

By Gregory Barber

June 20, 2025

Two new notions of infinity challenge a long-standing plan to define the mathematical universe.

Mathematicians probe the boundaries of logic by inventing new types of infinity.

Wei-An Jin/Quanta Magazine

Introduction

Last winter, at a meeting in the Finnish wilderness high above the Arctic Circle, a group of mathematicians gathered to contemplate the fate of a mathematical universe.

It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating the nature of two new notions of infinity. The consequences, Aguilera believed, were grand. “We just don’t know what they are yet,” he said.

Infinity, counterintuitively, comes in many shapes and sizes. This has been known since the 1870s, when the German mathematician Georg Cantor proved that the set of real numbers (all the numbers on the number line) is larger than the set of whole numbers, even though both sets are infinite. (The short version: No matter how you try to match real numbers to whole numbers, you’ll always end up with more real numbers.) The two sets, Cantor argued, represented entirely different flavors of infinity and therefore had profoundly different properties.

From there, Cantor constructed larger infinities, too. He took the set of real numbers, built a new set out of all of its subsets, then proved that this new set was bigger than the original set of real numbers. And when he took all the subsets of this new set, he got an even bigger set. In this way, he built infinitely many sets, each larger than the last. He referred to the different sizes of these infinite sets as cardinal numbers (not to be confused with the ordinary cardinals 1, 2, 3…).

Lucy Reading-Ikanda

Set theorists have continued to define cardinals that are far more exotic and difficult to describe than Cantor’s. In doing so, they’ve discovered something surprising: These “large cardinals” fall into a surprisingly neat hierarchy. They can be clearly defined in terms of size and complexity. Together, they form a massive tower of infinities that set theorists then use to probe the boundaries of what’s mathematically possible.

But the two new cardinals that Aguilera was pondering in the Arctic cold behaved oddly. He had recently constructed them, along with Joan Bagaria of the University of Barcelona and Philipp Lücke of the University of Hamburg, only to find that they didn’t quite fit into the usual hierarchy. Instead, they “exploded,” Aguilera said, creating a new class of infinities that their colleagues hadn’t bargained on — and implying that far more chaos abounds in mathematics than expected.

It’s a provocative claim. The prospect is, to some, exciting. “I love this paper,” said Toby Meadows, a logician and philosopher at the University of California, Irvine. “It seems like real progress — a really interesting insight that we didn’t have before.”

But it’s also difficult to really know whether the claim is true. That’s the nature of studying infinity. If mathematics is a tapestry sewn together by traditional assumptions that everyone agrees on, the higher reaches of the infinite are its tattered fringes. Set theorists working in these extreme areas operate in a space where the traditional axioms used to write mathematical proofs do not always apply, and where new axioms must be written — and often break down.

Up here, most questions are fundamentally unprovable, and uncertainty reigns. And so to some, the new cardinals don’t change anything. “I don’t buy it at all,” said Hugh Woodin, a set theorist at Harvard University who is currently leading the quest to fully define the mathematical universe. Woodin was Bagaria’s doctoral adviser 35 years ago and Aguilera’s in the 2010s. But his students are cutting their own path through infinity’s thickets. “Your children grow up and defy you,” Woodin said.

The Universes of Set Theory

Most mathematicians don’t concern themselves with these kinds of questions. They work with a set of nine assumptions, or axioms, about how sets behave, known as ZFC — “Zermelo-Fraenkel set theory with the axiom of choice.” These nine rules can’t be proved. Mathematicians have simply agreed that they provide a natural foundation for the rest of mathematics. From them, mathematicians build up rigorous proofs of all their conjectures.

Kurt Gödel showed that some reaches of mathematics will forever remain inaccessible.

Kurt Gödel Papers, the Shelby White and Leon Levy Archives Center, Institute for Advanced Study

But in 1931, the German mathematician Kurt Gödel demonstrated that any interesting system of mathematical axioms is doomed to incompleteness. There will always be true statements that can’t be proved. To prove those true statements, mathematicians would have to add a new axiom. But then this longer list of axioms would also lead to true but unprovable statements. And so on. Anyone who wants to prove all the possible statements in the mathematical universe will be forced to keep creating new axiomatic systems forever.

This means that the mathematical universe, which mathematicians often call V, is fundamentally unknowable. But set theorists want to describe it as closely as possible — to create model universes that resemble the real one while being easier to study. These models provide mathematicians with the extra axioms they need to prove those elusive statements about “smaller” axiomatic systems (like ZFC) while giving them confidence that the extra axioms they’re using aren’t arbitrary. “As you strengthen these theories, you end up making low-level mathematics more concrete. It firms up,” Meadows said.

Gödel provided a starting point. He built a model, which he called L, by beginning with the empty set (which is what it sounds like) and iteratively constructing bigger sets from there. The model was a good one and easy to work with, but it was also partial. It didn’t include the large cardinals — those stranger infinities that can’t be constructed using the same methods as Cantor’s. (L is therefore referred to as an “inner” model of V, because it lives inside the bigger universe.)

Set theorists aim to expand this picture. Throughout the 20th century, they defined more large cardinals, with names like strong, compact, supercompact and huge. Each new definition requires the creation of a new axiom; set theorists then hope to show that this new axiom is consistent with ZFC, that it doesn’t violate the most fundamental rules of mathematics.

These large cardinals also seem to form a surprisingly neat hierarchy, a true tower of infinities. Each large cardinal is much, much larger than the one below it, for example, and the axiom that defines it can be used to prove far more statements than the axioms that define lower cardinals. Moreover, due to this hierarchy, if mathematicians can show that one large cardinal is consistent with ZFC, it will imply the consistency of all the cardinals below it in the tower. “You might think, ‘Oh, it’s just going to be complete chaos.’ But it doesn’t seem to be,” Meadows said.

Whenever a new cardinal is added to the tower, proving its consistency also requires the development of a larger, more sophisticated inner model. “You put in there just the minimum number of things that are necessary so that in the final model, your large cardinal will exist,” Bagaria said. With each new model, the defined mathematical universe expands.

Hugh Woodin has an audacious plan to map V, the entire mathematical universe.

Courtesy of Hugh Woodin

For Woodin, the dream is to build an inner model that truly approximates V and therefore includes all the large cardinals. He calls it “Ultimate L.” It might seem like a hopeless task — after all, because of Gödel’s incompleteness results, it should require the construction of infinitely many inner models, each one containing yet another indescribably large cardinal.

But 20 years ago, Woodin discovered a shortcut: You don’t have to build inner models for all of the large cardinals. Reach a certain point in the hierarchy — the large cardinal called “supercompact” — and the model inherits all the large cardinals above it. “You get everything,” Woodin said. “Magic happens.”

But his plan to build Ultimate L relies on the mathematical universe being nicely structured, with the large cardinals forming a neat, hierarchical tower. In mathematical parlance, they must be “hereditarily ordinal definable,” or HOD.

Woodin mapped out two possibilities. “V is either very close to HOD or very far from HOD,” Bagaria said. “There’s no middle ground.” If you could find one thing that broke the hierarchical model, many other things likely would break it as well; chaos would reign. “Maybe the universe does contain many things that are not definable,” Aguilera said.

But Woodin conjectured that the first option — that V is nicely structured, with a definable tower of cardinals — is correct. So far, the evidence suggests he’s right. No one has been able to find a large cardinal that doesn’t fit into the tower while staying consistent with ZFC.

Now Aguilera and his collaborators are complicating the picture.

Order Versus Chaos

There’s some circumstantial evidence in favor of mathematical chaos. For one thing, despite many decades of work, mathematicians’ progress on the Ultimate L program has been slow. Woodin himself has experienced periods of doubt, though he currently believes that achieving Ultimate L is possible.

Set theorists have also identified very large cardinals that seem to break away from HOD, though they are extreme outliers. Defining these cardinals involves getting rid of one of the nine axioms of ZFC — the so-called axiom of choice. Throwing out a fundamental axiom is not, to many mathematicians, an appealing approach.

Juan Aguilera recently defined two new notions of infinity that, he claims, complicate our picture of how the mathematical universe might be structured.

Ulrich Zinell

That’s where Aguilera, Bagaria and Lücke entered the fray. In their recent work, they’ve produced two novel types of infinity, which they named exacting and ultraexacting cardinals. These cardinals, crucially, don’t violate the axiom of choice.

Defining them was relatively straightforward: They’re essentially bigger analogues of other types of large cardinals. But these new infinities exhibit strange properties. At first, they appear to fit into the usual hierarchy: It’s clear where in the tower they should live, in terms of size and complexity. But then the trio tried combining them with other, smaller cardinals.

Usually, when you add up large cardinals that are compatible with HOD — yes, it’s possible to add infinities to each other — you get a sum that’s roughly the same size as the largest cardinal in the mix. It’s not possible to jump to a higher part of the hierarchy. (Consider how, if you add 100 to a quintillion, the sum will be on the same order as a quintillion; you won’t end up with a much, much bigger number.)

But add a smaller cardinal to one of the new infinities, and “they kind of blow up,” Bagaria said. “This is a phenomenon that had never appeared before.” It implied the existence of a much, much larger cardinal — an infinity larger than anything mathematicians had imagined to be consistent with ZFC.

Joan Bagaria enjoys searching for new ways to expand mathematical reasoning.

Julia Bagaria

“This challenges the intuition we had of how large cardinals relate to each other and how they’re ordered,” Aguilera said.

“We have discovered this new region where the universe becomes so wild, so complicated, that it can no longer be equal to HOD,” Bagaria said. And there are probably lots of other bizarre infinities dwelling out there, waiting to be explored, he added.

According to Aguilera, the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things that we can’t see,” he said.

Mathematical Experiments

Still, this does not necessarily disprove the HOD conjecture, or render Woodin’s Ultimate L program moot. In fact, Woodin had previously considered a possible large cardinal that was, in retrospect, equivalent to the ultraexacting cardinals. But he discounted it because he did not think it was consistent with ZFC, which would make it irrelevant to our mathematical universe.

The team’s next step is to gather evidence to the contrary, to show that both of the cardinals they have defined are consistent.

But the math of set theory, which so often dwells in a world beyond proof, tends to look more like physics or biology, as Aguilera put it. You come up with a principle and test it experimentally; you see if it leads to contradictions, and analyze how it compares with things you already know. You might not be able to prove it true just yet, but with the right experiments, you can give good evidence for your claim.

Philipp Lücke’s most recent work on large cardinals showcases how the world of set theory often exists in a realm beyond mathematical proof.

Courtesy of Philipp Lücke

Aguilera, Bagaria and Lücke have provided evidence that their new cardinals are consistent with ZFC — that’s what the bulk of their new paper is about. For instance, they’ve shown that exacting and ultraexacting cardinals have similar structure and behavior to other large cardinals. Their definition also implies that if another, more established type of infinity is consistent with ZFC, then ultraexacting cardinals should also be consistent — and there’s been a lot of recent evidence for the consistency of that other type of infinity.

This, the trio argues, makes the new cardinals compelling candidates for showing that HOD is far from V, that the mathematical universe isn’t nicely ordered.

Not everyone is convinced. Gabe Goldberg, a set theorist at the University of California, Berkeley, points out that while the trio might feel confident in their evidence for consistency with ZFC, surprises frequently happen in the study of the infinite. Certain assumptions might no longer hold, causing the mathematicians’ arguments to crumble. It will take much more to truly refute the HOD conjecture, he said.

Related:

1. To Settle Infinity Dispute, a New Law of Logic

2. How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

3. How Big Is Infinity?

Woodin agreed. “I think we have to be very, very careful,” he said. He still thinks that HOD is close to V — and plans to continue his decades-long journey toward building a description of the mathematical universe that’s as complete as possible. He’s currently writing a “very long manuscript” on his latest approach to Ultimate L.

Other set theorists are excited by the potential of the new infinities. To them, chaos is the more exciting option. “I mean, in some ways … if Ultimate L worked out, it would kind of end a really interesting chapter in set theory,” Meadows said. “It would be like Alexander when he gets to India, finding there’s no more world to conquer.”

Perhaps there’s much more left for mathematicians to conquer. It’s time, Aguilera said, to explore what else is out there. “I always tell people that mathematics is infinite, but time is not,” he said.

2025 Jun 18

- How AI Models Are Helping to Understand — and Control — the Brain

Q&A | VIEW ON WEBSITE

How AI Models Are Helping to Understand — and Control — the Brain

By Eric James Beyer

June 18, 2025

Martin Schrimpf is crafting bespoke AI models that can induce control over high-level brain activity.

Martin Schrimpf wants to use AI to learn more about how human brains work.

Samuel Rubio for Quanta Magazine

Introduction

For Martin Schrimpf, the promise of artificial intelligence is not in the tasks it can accomplish. It’s in what AI might reveal about human intelligence.

He is working to build a “digital twin” of the brain using artificial neural networks — AI models loosely inspired by how neurons communicate with one another.

That end goal sounds almost ludicrously grand, but his approach is straightforward. First, he and his colleagues test people on tasks related to language or vision. Then they compare the observed behavior or brain activity to results from AI models built to do the same things. Finally, they use the data to fine-tune their models to create increasingly humanlike AI.

The process works best with more data and more models, so Schrimpf built an open-source platform called Brain-Score that contains nearly a hundred human neural and behavioral data sets. Researchers have tested thousands of AI models against the human data since Schrimpf first developed the platform in 2017, back when he was still in graduate school.

Schrimpf originally planned to work in the tech industry, but after co-founding a pair of software startups during his early academic career, he felt unfulfilled. “I thought I could ask neuroscientists how the brain works, and that would help me build better AI,” he said. “But I realized there’s a huge opportunity in the opposite direction: prototyping ideas in silico [on a computer] and using AI models to explain the brain.”

He moved from his native Germany to the U.S. to get a doctorate in brain and cognitive sciences at the Massachusetts Institute of Technology. In 2023, he moved back to Europe to start the NeuroAI Lab at the Swiss Federal Institute of Technology Lausanne.

Schrimpf’s experiments have shown that the brain’s visual and language systems process information in a similar way.

Samuel Rubio for Quanta Magazine

That year he also co-authored a study showcasing how AI models could transform neuroscience. Schrimpf and his colleagues trained a model to generate sentences that, when read, would activate or suppress neural activity in the reader’s brain. When they tested it with human subjects, brain scans confirmed that the AI-generated sentences really did alter neural activity in the way the model predicted. The study marked the first time that researchers in any field had exerted noninvasive control over high-level brain activity. Using this approach, scientists could potentially use AI-generated stimuli to help treat depression, dyslexia and other brain-related conditions.

Quanta spoke with Schrimpf about what artificial neural networks reveal about intelligence, the future of neuroscience, and the ethical considerations of predicting — and influencing — human thought. The interviews have been condensed and edited for clarity.

You study vision and language systems in the brain. Why these?

I want to build a model of the brain. Starting in vision was a practical decision because that field had produced most of the data in neuroscience, mainly because screens are good at showing many stimuli in rapid succession. Moving to language was a decision to see if the techniques we were developing for sensory systems, like vision, would translate.

Schrimpf on the campus of the Swiss Federal Institute of Technology Lausanne.

Samuel Rubio for Quanta Magazine

And did they?

Yes, it seems the language system in humans can be considered an encoder of features, just like the visual system. It might mean the way mental representations of words or objects are built in the brain is more widespread across cognitive systems than we assumed.

It’s famously hard to understand artificial neural networks, so when a model does seem to align with real neural data, how do you know it’s not just a superficial correlation?

We have all the information with these models. It’s just very difficult to parse. We try to let the data speak for itself. You could compare two human brains and find their activity patterns are similar. That’s basically what we’re doing for models with all the neural and behavioral data we have on the Brain-Score list.

If there’s a lot of data, and the models just keep approaching the ceiling — which I think is the situation we’re in for vision and language — then they might not be perfect, but they’re starting to be aligned.

How similar are these AI systems to the brain?

Artificial neural networks have a neuron-level similarity to the neuronal processing units in the brain. They can reflect activity that’s reasonably consistent with the brain and can even mimic human behavior.

Schrimpf holds his doctoral graduation hat from MIT, which includes a 3D model of his brain and other mementos of his time there.

Samuel Rubio for Quanta Magazine

We might never perfectly explain the brain with simplified models. But the much more interesting question to me is how useful the current models already are for brain science. And I think they are much better than most people give them credit for.

Some neuroscientists have said your approach doesn’t account for psychological data. Is that fair?

It’s true, in many ways, we’re throwing out the classic neuroscience approaches. We’re saying, “Let’s get more data and build models where we might not actually understand the internal mechanisms.”

Classical neuroscientists tend to react to our research with a mix of positive disbelief. I think many aren’t aware of how good these models already are at mimicking brain function.

I don’t see this as just one approach being correct. They both have their successes and limitations. It’s just different bets on which will be more effective in the end, and I’m betting on this AI modeling approach.

How close are we to a digital twin of the brain? Do you expect ever to see one?

That’s exactly what Brain-Score is trying to quantify! I’m optimistic we can get close, and I hope it only takes a few decades. If we get there, I’d think, “Cool, we did that. Now let’s see what we can do with it.”

Schrimpf has trained AI models to create sentences that precisely alter a reader’s brain activity.

Samuel Rubio for Quanta Magazine

And what would you do with it?

One of our dreams is to generate a font that will help people with dyslexia parse sentences. If we have a model of dyslexia, we can probe it and find changes to the text that make it easier for someone to read.

Or if we had a digital twin of a patient’s brain undergoing treatment for depression, we could optimize effective therapy. There’s also invasive stimulation — you could ask the model how to directly change the brain state into a less depressive one.

Are there any limits to influencing brain activity with AI?

In areas like decision-making or general memory, we’re still far from influencing neural activity. But if we can accurately model cognition, we should also be able to induce specific perceptual experiences we can measure.

This is an ethical minefield, though. How do you develop AI models that can responsibly influence thought?

We need to work with experts on that, and we’re exploring this as we move toward things that might become products someday. Creating legislative frameworks is critical, but it’s not obvious to policymakers what’s possible even today. There’s already a lot we can do with the brain that doesn’t have any kind of legal framework around it.

I do worry the timelines are going too fast. As we’re seeing with AI, by the time it gets to the public focus, there’s a lot of retroactive work to ensure everything is done properly. It seems that whatever society develops, security is an afterthought.

“Artificial neural networks have a neuron-level similarity to the neuronal processing units in the brain,” Schrimpf said. “They can reflect activity that’s reasonably consistent with the brain and can even mimic human behavior.”

Samuel Rubio for Quanta Magazine

You’ve found that AI models trained purely on computational tasks can still closely predict human neural responses. Does this imply that human intelligence is reducible to computation?

If you look at current AI systems, they look pretty darn intelligent. They have flaws, but I would certainly start to call the kinds of reasoning they are able to do intelligent. So, I do think that intelligence is reducible. We have a particular implementation that is biological, but we’re already seeing evidence that it’s not the only implementation.

Does that evidence change what you think it means to be human?

If we accept that human behavior arises from physical processes, then there’s no inherent limitation to building such processes artificially. AI models forgo biochemical synapses and use simple unit-level processing rather than complex cellular machinery. And yet, we’re seeing behavior emerge that is reminiscent of human cognition.

Related:

1. How Can AI ID a Cat? An Illustrated Guide.

2. Improving Deep Learning With a Little Help From Physics

3. Can AI Models Show Us How People Learn? Impossible Languages Point a Way.

So, I think the intelligence we see in humans is not exclusive to us. It’s a pattern of information processing that can arise elsewhere. Personally, I’m not unsettled by this. I view it as an opportunity to learn more about ourselves. What makes the human experience unique in my opinion is not the underlying building blocks, but rather the collection of experiences that are made in a lifetime.

Table of Contents

Quanta Magazine: 2025 Jun 26

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How AI Models Are Helping to Understand — and Control — the Brain