AI is solving some of the toughest math problems in the world, including ones that have stumped mathematicians for decades. What it means for human ingenuity when there are no more hard problems left to solve.

Guests

Steven Strogatz, professor for the Public Understanding of Science and Mathematics at Cornell University. Co-Host of the Podcast “The Joy of Why.” Author of the Forthcoming book Big Math.”

Also Featured

Hugo Duminil-Copin, Professor of Mathematics at the University of Geneva and winner of the Field Medal in 2022.

Francis Su, Math Professor at Harvey Mudd College and former president of the Mathematics Association of America.

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Transcript of Full Broadcast

The version of our broadcast available at the top of this page and via podcast apps is a condensed version of the full show. You can listen to the full, unedited broadcast here:

Part I   

MEGHNA CHAKRABARTI: When ChatGPT first came out in late 2022, math was not its strongest subject. Sure, it could write you a bad poem or draft a boilerplate email, but give it a middle school math problem, and there was a decent chance it would get the answer wrong. In one study from early 2023, researchers asked ChatGPT for the answers to a thousand algebra word problems.

It got 84% of them wrong. Four years later, everything has changed.

SAM ALTMAN: Three summers ago, our models were okay at grade school math. Two summers ago, they were pretty good at high school math competitions. Last summer, we reached a gold level medal in the most prestigious international math competition.

CHAKRABARTI: That is OpenAI CEO Sam Altman speaking at the United Nations just last month. He and Anthropic CEO Dario Amodei were briefing the UN Security Council about the accelerating capabilities and risks of artificial intelligence, and both of them pointed to math as evidence of the ever-increasing power of their models.

Four years ago, it could barely write a line of code or solve a high school math problem. Today, it writes most of the code at Anthropic and solves world-famous unsolved math problems.

And just a few weeks ago this summer, one of our models solved one of the Millennium Prize problems, the Navier-Stokes equations. We are not solving math problems for their own sake, but to empower people to discover new knowledge.

The trajectory only needs to continue for a tiny period longer, one or two years, maybe less, to reach what I’ve called a country of geniuses in the data center.

CHAKRABARTI: The Navier-Stokes problem is one of the great unsolved problems in mathematics, or rather it was.

When OpenAI announced its model had solved one of these problems, or the great problem, it was a shocking transformative moment in both math and in AI. But we should note that the solution that OpenAI has presented is still being evaluated. It turns out that OpenAI, though, was just getting started. Last week, the company released more than 700 mathematical papers claiming new results on hundreds of problems that some of the world’s greatest mathematicians have spent decades trying to solve.

One of those mathematicians is Hugo Duminil-Copin at the University of Geneva.

HUGO DUMINIL-COPIN: I see mathematics both as a language for science and a very powerful tool for science, but I also see it as a study of humans themselves. How do they think? How do they abstract? How do they model the world around them?

CHAKRABARTI: In 2022, Duminil-Copin won the Fields Medal. It’s sometimes called the Nobel Prize for mathematics. He has spent his career pursuing some of the most difficult problems in mathematical physics, and he compares those great unsolved problems to lighthouses in an unexplored landscape.

If you are an explorer and you have this huge landscape in front of you, you have to set some goals, some places that you want to discover more than others, some direction that you want to pursue. And that’s why I use the idea of the lighthouse. Because that’s the direction to go home, to go to the places that matter.

CHAKRABARTI: OpenAI’s data dump last week just annihilated all of Duminil-Copin’s lighthouses because the company claims to have found solutions to 16 questions connected directly to his research.

DUMINIL-COPIN: It just killed, bulldozered all the problems I was working on. I was already very well aware that now the artificial intelligence frontier models, they are so strong that they can achieve things you cannot achieve. I really sincerely didn’t think that it would be so extreme that they all disappear at once.

CHAKRABARTI: Duminil-Copin is now working to understand how OpenAI arrived at the answers. He still is working, teaching students, but right now he has no idea what to research next.

We are the day after tomorrow. We need to rethink our way of thinking because simply all the lighthouses shut down.

CHAKRABARTI: Consider this. OpenAI says its artificial intelligence took just 88 hours to find a solution to the Navier-Stokes equations. They unleashed 10,000 bots and spent an estimated $15 to $20 million just to solve this math problem. But here’s where I think it’s really important, because this is not just a question of the future professional viability of a tiny handful of the world’s most brilliant human mathematicians.

There is something profound and deeper at stake. Like Hugo said, math is the science of art and abstraction. It is the art of abstraction, and as such, it is also an expression of human creativity, of the process of thinking and working hard to create new ways of seeing and understanding the world. And in that way, I think it’s like art, writing, music-making, even trying to think just creatively about how to solve everyday problems in our own lives.

So I would argue that what’s at stake is what we could potentially lose when AI short-circuits the necessary hard work of all creativity. What happens when there’s no hard problems left to solve? Let’s turn for answers to Steven Strogatz. He’s a math professor at Cornell University and also a professor for the Public Understanding of Science and Mathematics.

He hosts the podcast The Joy of Why and is author of the forthcoming book Big Math. Professor Strogatz, welcome to On Point.

STEVEN STROGATZ: Hi there, Meghna. It’s great to be with you. Thanks for having me.

CHAKRABARTI: Okay. So first of all, give me your take, your reaction when you heard about, not just one, but several of the most intractable problems of mathematics being slayed by AI models.

STROGATZ: Oh, I really appreciate the way you opened the show with, Hugo Duminil-Copin. I had only, I’ve never met him. I read about what was happening to his life’s work, really. That’s the way we should put it. He has these series of problems that took actually great creativity just to propose.

We might think the solution is the part where the creativity happens, but your listeners should appreciate that coming up with a good puzzle is actually a very tricky thing. If a puzzle is too easy, that’s not fun, that’s not interesting, it’s not generative, it doesn’t lead anywhere. But if a puzzle is too hard and it’s intractable, that’s also fruitless. So hitting that sweet spot takes a great deal of taste and artistry and Hugo is one of the great mathematicians, great human mathematicians we need to say now.

So it’s been devastating for the part of math that we refer to as pure math. And I hope we’ll get into this distinction between pure and applied because a lot of folks are reacting in public, like when I’m on X or BlueSky, I see a lot of the public seeming confused “Don’t you want your problem solved? What are you guys complaining about?” So I hope we’ll get into that too.

CHAKRABARTI: Oh, yeah. That’s the whole point of this hour, Steven, so don’t worry.

STROGATZ: Good. Okay.

CHAKRABARTI: But for folks who aren’t familiar with the, the world of, this highest level of mathematics, can you just give us some facts here?

Like the Navier-Stokes problem or equations, what was that? Is that?

STROGATZ: Sure. Yeah. Okay, let’s start with Navier and Stokes are scientists from the 1800s, like the middle 1800s. They’re interested, or they were interested in trying to explain things that we see all around us. Imagine you’re in your backyard, leaves swirling around.

That’s the flow of air causing a little tornado, a little vortex. So Navier-Stokes equations are all about the flow of air or water or anything else that can flow. It could be blood, could be lava, so there’s a lot of things in nature that involve flow. When we have, if we’re designing a car and we want it to be optimally streamlined, the engineers could put the car in a wind tunnel, but then you have to actually build the car first.

So nowadays what we do is a much cheaper and in a way, more efficient thing, which is to put a computer model of the car through the same kind of paces, blow computer wind over a computer car by solving the Navier-Stokes equations in the computer, and then you can change the design of the car very quickly to see, if you want to make some adjustments.

Same thing with airplane design or ship design, or if I’m in a, like I’m an interior architect and I want to think about a ventilation system in a room. Yeah. You have to think about it. So Navier-Stokes equations are all about airflow and water flow, would be the short answer.

CHAKRABARTI: Okay, and then what made them so hard to solve?

I’m still trying to understand why this caused, OpenAI’s announcement caused such a earthquake in mathematics. Okay, we should distinguish two things that are hard about them. That one is what’s hard in solving them, the other is what was the actual millennium problem because it’s a misnomer to think that it’s about solving the Navier-Stokes equation. Okay. So let’s take the first part first because that’s the part that really matters in the real world. The difficulty is that air can be turbulent, and so can water. Anyone who’s flown across the country or across the ocean knows about this, that air can get choppy and bumpy.

And what happens is if you watch the way that the streamlines go, like if you watch, or I mean you can feel it. Maybe a little more direct experience would be with your faucet in your sink. If you turn on the faucet but only open the tap a little bit, the water comes out in nice smooth sheets.

That’s what we call laminar flow, meaning sheets. Like it looks calm, but as you start opening up the faucet more and more, it starts to get really burbly and wild and picture the froth on an ocean. Same thing is happening in the atmosphere when you’re on a bumpy plane ride, that instead of the wind or the air moving smoothly, which is a nice smooth ride, if the air starts making these little vortices, little tornadoes, like the swirling leaves in your backyard, but at the scale of a whole airplane, you’re going to get bumped all over the place.

Okay. So the difficulty, what the Navier-Stokes equations do a fantastic job of predicting is turbulence, motion of air associated with climate or weather patterns. So we use it all over the place, but they’re really hard to understand because they have a bad or a nasty mathematical property in which the whole is not just the sum of the parts.

Does that little phrase make sense to you? Like here’s why that’s so bad. You can’t break the problem into pieces. That’s the way we do science.

CHAKRABARTI: I was gonna say, that’s a fundamental for how you’re supposed to approach problem-solving.

STROGATZ: See, you get it. That’s the point, that science proceeds by this thing we call reductionism, where you take a hard problem, break it into little problems, solve the little problems, and put them back together, and that doesn’t work with water or air or Navier-Stokes.

Part II

CHAKRABARTI: Okay, so Steven Strogatz, tell me about the millennium is a word I can actually say, believe it or not, but the millennium problem aspect of the Navier-Stokes equations.

STROGATZ: Sure. Yeah. So this is when we enter the realm of pure math as opposed to applied math.

So with applied, that’s where we’re talking about real blood or real water or real flow of air. But when mathematicians think about the Navier-Stokes equations, these very idealized, pristine description of … I don’t want to say it’s real water anymore because it leaves out a lot. For instance, the main thing it leaves out is that we know real water is made of water molecules, H2O molecules. The Navier-Stokes equations pretend that’s not true, not because they never took a science class, it’s more if you want to analyze the flow of something made up of gazillions of molecules, that turns out to be extremely difficult, and it’s easier, believe it or not, if you imagine it’s made of infinitely many rather than trillions.

You might think infinity is harder than trillions, but actually infinity is easier because we get to use calculus.

CHAKRABARTI: I was going to say, I’m seeing limits here now in my head all of a sudden.

STROGATZ: There you go. That’s why we put you through all that stuff. The stuff about limits and infinity and infinite sums and integrals and derivatives for your listeners, whoever were in a calculus class, it’s because infinity is your friend in math.

If you can harness the power of infinity by, let’s say, taking an idealized picture of water and imagine breaking it into smaller and smaller bits, and you go all the way down to the limit, like there’s no such thing as atoms or molecules in idealized water. It can be arbitrarily finely cut up.

That lets you do calculus on it, and in fact, the Navier-Stokes equations are calculus equations of a type called differential equations. Okay. So one thing we know is that the Navier-Stokes equations can’t work once you get down to the molecular scale. We know that they’ll break down there. But what the millennium problem is about is in math, we often like to do wishful thinking.

We say, “Look, we know that they’re going to screw up at the atomic scale or at the molecular scale,” but still, the Navier-Stokes equations give us so much information across many scales of interest. We as people are not molecular scale either. They describe what’s happening in our lives very well.

So we want to understand their mathematical properties better, and here’s where we get to what the question was. The question was, do these equations have anything else fundamentally wrong with them besides this lack of atomic aspect? And in particular, if we let them evolve, that is we solve them, we let them predict what the flow of our hypothetical air or water will be, could they ever suddenly produce what’s called a singularity?

Meaning someplace where the flow becomes infinitely fast or infinitely spinny, like it could spin in an infinitely strong tornado. Could that ever happen in the math? Not in the real world, could that happen in the math? And if it could, that would be considered a serious pathology of the equations, and we just didn’t know because of this aspect that I mentioned that we can’t break the equations into parts to analyze them. We just didn’t know, and so now we do know, thanks to the work of OpenAI’s 10,000 agents. They showed what was actually, to some people, surprising, though I think most mathematicians would’ve thought this was the way it would go, that in fact, the Navier-Stokes equations do have this pathology in them.

CHAKRABARTI: I have to say, I love how mathematicians talk about problems. Pathology in the equation. But this is actually, this is a great way to return to what I think is the sort of fundamental challenge before us as AI solves more and more hard problems, and not just in mathematics. So Steven, that’s what I’d like to really dive in deep on with you.

And I want to just note to everybody that we did reach out to OpenAI and to Anthropic to get someone to come on or comment from them about various questions that we have. They did not respond, but on the other hand,Sébastien Bubeck, who is a researcher at OpenAI, and he led a lot of the company’s work around math involvement, he has talked on an episode of OpenAI’s own podcast in April of this year.

BUBECK: The nice thing about mathematics is that the question are very clear, non-ambiguous. Everybody agrees on what the question is asking. So that’s point number one. Point number two, you can verify the answer. So once a model can give an answer, everybody will agree was it correct or was it not correct. Mathematics was just the perfect benchmark to see the model making progress during the last four years.

CHAKRABARTI: By the way, we reached out to Sébastien Bubeck personally, and as, as I said, also OpenAI as an organization, did not hear back. Interesting that he says, “Oh, math is actually a great benchmark to see how good the models are getting.” In the process of making that measurement, which works for OpenAI, something else is getting lost.

Francis Su is a math professor at Harvey Mudd College in California and a former president of the Mathematics Association of America. He says math is more than just those numbers, equations, and variables. It is also a set of virtues.

SU: It’s a set of character qualities, habits of mind, dispositions that enable you to live in the world with joy and wonder and awe and responsibility. If we lose sight of that, then of course people are going to have this idea that math is just calculation, and I don’t want people to think of math that way, right? Math is more than that. Math is understanding. That’s the whole reason anybody’s ever been involved in the mathematical pursuits, to try to deeply understand things, and we can’t cede that to AI.

CHAKRABARTI: So Professor Strogatz, when you mentioned calculus as one of the things at the heart of the Navier-Stokes equation, that got me thinking too, if you don’t mind, I’ll bring in Sir Isaac Newton into this. I think it’s like a … he’s a perfect example of how when mathematicians find themselves at the limit of their understanding, something fundamental, creative, like purely human and creative happens to create new sets of equations or even entirely new forms of mathematics to help solve the problems that they’re asking, and that’s essentially what happened, I think with Newton, right?

And so can I ask you to describe, if this is even possible, what the creative process is for mathematicians when they’re grappling with a problem, or even what question to ask?

STROGATZ: Great. I love it. Yeah. Excuse me, I have so many reactions to the clips that you just played, but let’s go with Isaac Newton.

You can’t do better than Isaac Newton. So just to remind people of a little bit of the history, that people had been looking at the planets forever. They’re the wandering stars they used to be called. If you look, if you’re a person in a society where there’s no artificial light, you’re out at night, it’s dark, it’s magnificent, you’re looking at the sky.

You see the stars, you learn about constellations, but then you see these odd wandering stars that are not in the same place every night relative to the other stars. They wander. They actually move across the sky as the seasons change. This is what we call the planets, Venus and all the rest of them. And forever, people wanted to understand what are the patterns of planetary motion.

Now, Johannes Kepler, before Newton, figured out that there, you could actually see mathematical patterns in the motions of the planets, and he wrote down three of these famous Kepler’s laws. But what was so spooky is that math appeared to be governing the motions of the heavens, but he couldn’t explain why.

Kepler was actually a very religious person. He was training to be a minister, but he turned out to become an astronomer and a mathematician instead. Excuse me. And he had no explanation for the why did the planets move in ellipses around the sun. Why did all the planets have the sun as a focus of the ellipse, a certain special point in the ellipse?

Excuse me. So Kepler had some idea that the sun was special and that there must be some kind of mysterious thing emanating from the sun that was holding the planets, but he didn’t know what it was. Now, fast-forward a few decades and you get Isaac Newton, who says he imagines that there’s some kind of force.

Today, of course, we would think of it as the force of gravity. And people were aware of gravity. They knew things fell on the Earth. But it didn’t occur to them that the same force might be holding all the planets to the sun. It’s an invisible force. Anyway, Newton wrote down math to describe that force and the motion of the planets, and he could predict everything that Kepler saw.

So we consider this a triumph of human understanding, that with calculus he could actually derive everything Kepler saw from logic, which was a really uncanny thing. It’s like nature knows the same logic as this primate species that we happen to be.

CHAKRABARTI: But it took … used the word imagine, right?

STROGATZ: He did imagine.

CHAKRABARIT: It took a leap of imagination, right? Because there’s a huge bridge that Newton crosses mentally when he says there’s a force that’s holding all of these planets together, right?

STROGATZ: Let’s just think about that in case people are brainwashed that, oh, it’s just the force of gravity.

Think of how weird that is. In fact, Newton himself commented very much, not only how weird, but in fact how wrong it was. Because it’s not like there’s a rubber band connecting Jupiter to the Sun. You know what I mean? This is an invisible thing, and Newton in his theory said the Sun pulls on Jupiter instantaneously, and Newton knew that didn’t make sense because how are you going to get from the Sun, whatever it is that’s pulling, how does it reach from the Sun all the way out to Jupiter or the Earth instantly? He thought that wasn’t right, but he said, and this is such a brave thing that scientists and mathematicians do all the time. He said, “Let’s just pretend it’s instantaneous. What do I derive using logic from that assumption?” And he could actually derive all of Kepler’s principles.

CHAKRABARTI: Okay. So I have accidentally happened upon what I think is a really great example with Newton, so let’s stick with him for a little bit longer. What else can you point to that emerge in terms of our broader scientific understanding about the natural world from Newton’s leap of imagination regarding there’s a force?

What else did we learn?

STROGATZ: Everyone of my age, maybe not young people like you, but look, we got to the Moon and back. That was a very heroic thing, and it was all based on Newtonian physics. But, along the way, in case people are thinking okay, was it really worth going to the Moon?”

Let me just mention that in trying to do something hard, because this actually bears on our broad discussion here today. In trying to do something hard, the engineers at the NASA Instrumentation Lab had to invent an onboard computer that could sit in the lunar module, that could guide Neil Armstrong and Buzz Aldrin down to the surface of the moon safely.

And this was at a time when computers would occupy an entire room. They had to make a computer that could fit in a suitcase. In other words, they invented the personal computer long before Steve Jobs and Steve Wozniak, because they had to solve this hard problem of getting someone to the moon safely and back.

So I’m just saying that when you try to do something hard, using Newtonian physics and engineering and courage and daring and aviation, all of that, you might have a spinoff like the personal computer.

CHAKRABARTI: Yeah. Yeah. In fact, I would say that not just in math and science, but in everything that people do, right?

When they have to try hard, there are always spinoffs. That’s why we have the aphorism about, it’s about the journey, not the destination. Because, and returning back to science, or actually not any discipline as people share the struggles they had in that journey, right?

In science and math, it’s paper publication. In art it’s talking about your artistic process or showing versions of a project that didn’t exactly work. In writing, it’s draft after draft. When they share the struggles … I would say very often, those struggles actually inspire other people who go down a different path and continue the expansion of human understanding.

STROGATZ: True. True, and the false starts are often very enlightening. We learn a lot from our mistakes as well as our successes, so I like that you’re putting an emphasis on trial and error.

CHAKRABARTI: That’s just one aspect of how human creativity works.

But do you want to, can you hypothesize with me for a second, Steven? And you can tell me this is an utterly ridiculous question to even ask, but I’m into doing those kind of questions. What if in the 18th century, Sir Isaac Newton could just plug in a question into a computer, and be like, “Why is it that these planets rotate around in the sky in this really predictable way?” What do you think we would have lost?

STROGATZ: That’s great that we’re going there because this is such a rich question. Let’s see how much we can get through. Understanding, number one, if you just had answers without understanding, you would lose so much power to do things that really have benefited humanity and society.

Not only it’s difficult to even begin talking about this because there are so many. But let’s take Newton’s great successor, Albert Einstein. So he was very profoundly moved by the harmony of nature, by the mathematical structure of the laws of the universe and all that, but in thinking about how to improve Newton’s theory of gravity, he gave us a theory that now lets us have these GPS gadgets in our car or in our phone.

So what I’m trying to say is that by pursuing understanding for its own sake, you have spinoffs that have really just improved our daily lives. That I would call the instrumental value of understanding. If you just had answers, you don’t get the instrumental value of understanding. The second thing is that understanding feels good.

There’s an experiential value of understanding in that anyone who does puzzles knows it’s fun to have the aha moment, the eureka moment. And you could say who cares if it’s fun? That’s frivolous. I think it’s one aspect of what makes life worth living, that joy is actually useful.

What else are you going to do while you’re here? Wouldn’t it be nice to be happy? So that’s number two. I could give you more, but let me pause and see if you wanna jump in with something.

CHAKRABARTI: Oh, all I was gonna say is you’re speaking to my soul right now, Steven. Because I’m the coach of my son’s LEGO robotics team, and he and I have spent the past three days absolutely struggling over a particular design that he wants to do for a lever arm. And there has been a lot of struggle. There’s been moments of outright rage between the two of us.

STROGATZ: Okay. Rage, good. That’s a human emotion.

CHAKRABARTI: But we have made multiple breakthroughs in terms of, as you said earlier, every design that he’s tried that hasn’t worked, that has allowed us to figure out, okay it doesn’t work for these basic laws of physics, so but how do we modify that?

And then I think yesterday we might have had something of a breakthrough. Can’t say for sure yet but it was a thrill. It was a thrill.

STROGATZ: Okay, come on. Let’s talk about all the great things in your story. The thrill, the rage, because remember, rage and intense emotions, we know from neurobiology that those help us form memories.

When you have an intense emotion, like, you and your son are gonna remember that time you helped him with the LEGO robot stuff, and that’s gonna be precious to you as he gets older, as you get older, and if you just had the answer, you would’ve missed all of that.

Part III

CHAKRABARTI: Steven, as I mentioned earlier, we did reach out to OpenAI. They didn’t get back to us, but we’ll hear from one of their researchers again in just a second. But I want to turn back to Francis Su, the math professor at Harvey Mudd College, and what he doesn’t like, of the many things, is the way that OpenAI released all of their findings basically at once and with minimal collaboration with other mathematicians.

SU: It’s like a show of domination. It’s like a show of saying, “Hey, look, here’s our latest toy, this unreleased model that nobody has access to, and look what it can do.” But it’s not thinking about the impact or effect that this might have on the development of mathematics or the development of mathematicians.

CHAKRABARTI: As I mentioned earlier, we reached out to OpenAI. They did not respond to our request for comment. But in an episode of their podcast earlier this year, back in April, researcher Sébastien Bubeck said all this worry about solving problems is actually misplaced. There’s something else more important that motivates him.

BUBECK: We’re not solving problems to write papers, to show, to say that we can write 10 times more papers than our neighbor. That’s not the point. You can do competitive chess if that’s your kind of deal. We’re trying to really understand deeper things. And why are we trying to understand deeper things?

Because we want to have better control over our environment. We want to be able to cure diseases. We want to be able to build things, better, faster, more robust, more solid, all of those things.

CHAKRABARTI: Steven, if I understand correctly, do you know some of the mathematicians working either at OpenAI or in the AI world?

And if you do, how are they thinking about the work they’re doing?

STROGATZ: I do know some of them. In fact, I spoke to Sébastien just the other day before they dropped the bomb of those 300-plus problems and 700-plus manuscripts. And I don’t know him personally. I have not met him. But I wanna sorta complicate the story, the narrative that we’re telling here, because I think there is another side to this, and I want to be fair about it, which is that first of all, okay, I could be wrong, but my impression is that he is like a lot of the people I know. He loves math, too, Sébastien, and his colleagues really are sincere, I think, when they say they care about the young people.

That is, they’re not trying to destroy a generation of the next version of your son who want to go into a career in STEM, science, technology, engineering, and math. They, in fact, think they’re giving the world fantastically powerful new tools, and we can approach harder problems with them.

And why the negativity? Let’s get excited. There’s so much we can do. And just to complicate it a little further, we’ve been talking so far in our discussion about the importance of human understanding, but there’s an eerie thing that I want your listeners to take seriously, which is that AI can give us a new route to power without human understanding that has to be taken seriously.

And if I could, I just want to tell you a little story about that.

CHAKRABARTI: Absolutely.

STROGATZ: Can I spend a minute?

CHAKRABARTI: Please.

STROGATZ: Okay. It’s a story that I heard while working on the book that you kindly mentioned that I’m working on with my co-author, Alex Townsend in which we interviewed a woman named Regina Barzilay, who at age, well, let’s see.

I think she would’ve been 40, 41. She had her first mammogram, and she went in every year. It was at the time, the American Cancer Society said, “You should get annual mammograms starting at 40,” and she did. She had no risk factors that she knew of and no family history of breast cancer, so she was just going in the way you’re supposed to.

And on her third mammogram, bad news was given to her that they could see clear evidence of what seemed to be breast cancer, and she went through, and a real ordeal over the next, this was in April of 2014 that she heard the bad news. She had two lumpectomies. She went through chemotherapy.

All her hair fell out, she told me. She went through radiation treatment. It was terrible. Obviously terrible. She is alive, okay? … The good news is she’s alive, and she’s still talking to us. But at the time, she was a recent immigrant to the U.S. She had no family nearby.

She had gotten divorced recently. She had a little seven-year-old son. So you’ll be able to relate to this story. I’m sure all your listeners can too. And she was really scared and said, “If something happens to me, what’s gonna happen to my son?” Okay, so it happens that this person, Regina Barzilay, is a very brilliant computer scientist who works on machine learning and AI.

And she talked to the doctors at the place where she was being treated, in fact, in your place. Are you in Boston?

CHAKRABARTI: Yes. Yeah. We’re located in one of the medical meccas of the United States.

STROGATZ: Okay, so she is a computer science professor at MIT, and she was being treated at Mass General Hospital, one of the world’s best hospitals.

And she said to the doctors there I could actually help you with your data. I can do data science for you. Do you, by the way, do you know what my odds are for a woman with my prognosis, or sorry, with my diagnosis? What’s gonna happen to me? And they said we don’t really know.

CHAKRABARTI: I was gonna say, that’s a hard question for a doctor to answer.

STROGATZ: Yeah. They don’t really know.

CHAKRABARTI: They don’t wanna answer it either.

STROGATZ: Okay, it’s a tough question. Fair enough. But she said could I get … she wondered, because sometimes cancers can be slow-growing. Is it possible that there was something on her mammogram in the first mammogram, and maybe it could’ve been caught earlier, and the treatment would’ve had better chance of, you get the idea. It’s always better to catch it earlier. Okay. So anyway, with a team of radiologists and other computer scientists, she used AI to look at about a quarter of a million screening mammograms that had been collected of women where we knew the outcomes of, did they get cancer a year later after that mammogram, or two years or three years?

They have now developed the state-of-the-art, I would say the world’s best breast cancer risk prediction system using AI. And here’s the part, this is … Okay, so that’s the setup to the story. Now, here’s the interesting part, I think. That, of course, it’s fantastic they have created this. But when I asked her, “Do you understand what it’s seeing?”

When it makes a prediction, and she said, “No, it can’t tell us, and we don’t know.” And I said, “Doesn’t that worry you?” Because I’ve been brought up that understanding is critical. You don’t understand why it’s making the prediction it is. And she said this thing that I can’t get out of my head, which is, “Why would I limit the model to what humans can understand?”

Okay? Think about that. And we do this all the time. We have if anyone has used Google Translate, you know how valuable that is, how it can give you one language into another very fast, very idiomatic. That uses AI. That was based on machine learning, and if you ask it or the engineers who helped make it, what has it taught us about linguistics?

The answer is nothing. It just works. We don’t know why it works, but it works. So I want to make the point that we’re entering an era where understanding can legitimately be discussed. Is it as valuable as it used to be? It used to be a human route to power, but can we get to progress without it?

And if so, what do we lose?

CHAKRABARTI: I think that … so this is such a great question, and I think that I completely see Regina’s point of view here, right? Because is it necessary to understand the intricacy of how the model works if it’s providing life-saving information? Possibly not, right?

If you’re in the world of practicality, I would say no. We just want more people to survive cancer.

STROGATZ: Yes, that’s right.

CHAKRABARTI: But on the other hand, I would ask her in return, maybe I will call, we’ll call her up —

STROGATZ: You should talk to her. She’s fantastic.

CHAKRABARTI: But I would ask her in return … we don’t necessarily have to understand if the goal is to cure more cancers, we don’t have to understand exactly how the model works.

But if the goal is also to be able to know what to do when the model makes a mistake, maybe we should understand how it’s working. Or if the goal is to be sure that the model is ethically applied, maybe we should understand how it’s working. It becomes more a problem of humanity and less of predictability.

Do you see what I’m saying?

STROGATZ: I totally do, and this is why it’s such a pleasure to be on your show because you see everything. Sorry. No, really. You see this is when people are on the internet looking at the discussion happening about math and AI right now, it’s so shallow.

So please, people, listen to On Point, because this is where we go deep. No, this is a really not an easy discussion. So let me add one layer of complication to this that I hope will be reassuring, which is if you ask Regina, “Hold on now, you don’t understand how the thing works. How are you being responsible?”

She’s, of course, a very meticulous scientist, and they have done tons of statistical validation. They’ve tested it in countries all over the world, women of different races, different nationalities, everything you could think of. So there’s a lot of statistical evidence that it works. That’s not the same as understanding.

But this is the standard in medicine, where medicine is much more difficult than math. Math, like Sebastian said in your clip, math is easy compared to medicine because in math we know the right or wrong answer. It can be verified. In medicine, it’s way harder, and so people like at the FDA, think about how FDA’s standard works.

They say a drug has to be shown to be safe and effective. We do not need to understand its mechanism of action. That’s not required, okay? So doctors are used to this. They have to save lives and do it fast. They don’t have time like mathematicians, where we’re looking at books from 100 years ago.

We take our time. It’s a different, it’s a high stakes … so this is the question, like you raised, in these high-stakes settings. Now, you raised a fantastic point about trust. How do I trust something I don’t understand? When the stakes are high, like in medicine. What about finance? What about when everyone has AI advisors making stock decisions for them?

But you don’t know why. Are you going to trust that?

CHAKRABARTI: I think, and this is where Regina’s right in asking should we limit the models based on the limits of human understanding? No. But these models are also working in a world for people, and so what we have to do is find a way to maybe expand human understanding to get closer to what the models can do.

Now, I don’t know how we even begin to do that, but the other argument may be in favor of AI has to do with that latter part I talked about expanding the limits of human understanding. You heard Bubeck say this earlier. With AI solving really hard problems that just opens up more possible questions for us to ask to solve. That seems to be a compelling argument in favor of AI.

STROGATZ: That’s why we have to open this, we have to approach these questions in good faith. And be rational and try to be as calm as we can be, but also be very sensitive to the dangers here. This is a really precarious, high-stakes thing that, we’re doing a big experiment with our whole world, with all of human life and the life of the planet. So we wanna get this right. All of us, you know, whether we call ourselves techno-optimists or even doomers, we better get this right. And these are very difficult questions that are being raised, and we should stop calling each other names and try to really think about this and be calm and figure it out.

CHAKRABARTI: Also, I wonder, I don’t think we’re at the point, Steven or correct me if I’m wrong, where I think we’re in real trouble when it’s the AI itself that’s asking a new level of questions that people haven’t come up with.

STROGATZ: That’s a good point. Let’s be real about what’s happening right now and what’s not happening. Like in the math that we’ve been talking about earlier in this show, yes, the AIs are now superhuman at problem-solving. They are subhuman at asking good questions. They are subhuman at taste.

They don’t know what’s interesting. If you read their papers that they write in their attempts to solve these problems, they are absolutely abysmal explainers. However, they’re interactive. If I ask my $20 a month subscription of ChatGPT to explain what the internal model figured out, that thing can talk to me really well.

And I can say, “No hold on. Give it to me at a lower level. Pretend I’m 10 years old.” It can do that. So you can, as everyone knows who’s played with these things, you can dial them up and down, and they can be a very good teachers.

CHAKRABARTI: We’ve only got a couple minutes left, Steven, and I wanna return the focus of the conversation to actually something that you mentioned earlier about young people, right?

The next generation of humans, not just who are interested in STEM, but just being alive. You’re there at a university. There are lots of young people listening now, and my concern continues to be that for people to make progress themselves, and therefore for each other, like it or not we have to struggle.

The creative struggle is at the core of progress for Homo sapien. And forget millennium-level problems, but if we’re entering a world where people just have to struggle less to get to answers, therefore I don’t think we’re at risk of atrophying our creative muscles, aren’t we?

STROGATZ: Yes, we are. Sure. But I would I think I would push slightly back on you here. Push back hard if you want. Go ahead.

STROGATZ: Well, I don’t think the emphasis should be on creativity. I think I would wanna take it more towards Francis Su and his virtues. Because let’s talk about your son, unless it’s too personal, but I like to think about him, or I think about the students in my classrooms.

They’re 18, 20, 22 years old. When there’s nothing for them to do because the AIs can do everything, including all the creative part, all the asking the questions as well as solving them. If your son doesn’t need to struggle because the AI can build the Lego robot, suppose there’s a robot that builds the robot.

That’s not going to be that far off. That will happen, and he can just, your son can lie back on his chaise lounge like in the movie Wall-E. And do nothing. Just have his credit card hooked up to the internet. That’s all he needs to do. What’s going to be lost is, yes, human creativity, but what’s also going to be lost is something like dignity.

And what Francis Su is talking about is that struggle makes us better. We learn how to, we learn how to be better people in our lives.

This article was originally published on WBUR.org.