Math and Beauty in the Age of AI
20:46 Share

Count Socrates and Isaac Newton, among the great thinkers who had doubts about new ideas and new technologies, Socrates thought writing would sap our mental abilities to remember, think, and reason. And Newton, he thought that algebra was too abstract and didn't have the same intuitive power that geometry did.

But ideas in mathematics have evolved. Like calculators came along, made it easy to do arithmetic. Computers made certain kinds of problems almost trivial to solve. Math seems to be, at another inflection point, just as impactful as when writing upended the Socratic approach.

Welcome to the Quanta podcast, where we explore the frontiers of fundamental science and math. I'm Samir Patel, editor-in-chief of Quanta. In this episode, we're going to explore the landscape of modern mathematics to attempt to get a handle on how AI and other technologies, which are already being used in writing proofs and solving problems, stands to change research mathematics as we know it.

We recently explored this idea as part of our special issue, Science, Promise, and Peril in the Age of AI. One of the stories in it was written by our math editor, Jordana Sapelowitz, and she joins us today to talk about her piece that tackles how mathematicians are seeing this coming shift and what it means for the grand project of math in the coming decades. Jordana, welcome. Thanks for having me.

We like to, before we jump into these conversations, get a sense of where we're going. So what's the big idea that we're exploring here? Yeah, so the question kind of started as, if AI were to get really, really good at math, will there be a need for mathematicians? And while mathematicians certainly believe that they're not going to become extinct in

They now have to assess how AI will impact mathematics, how it will change what they do, what they value, and what mathematics really is at its core.

When we think about research mathematics, the mathematics that mathematicians are doing every day, it's not necessarily the popular conception of what math is. So can we make some distinctions here between arithmetic, algebra, and what we're thinking of as research mathematics? Yeah, I think when a lot of us hear the word math, we think about the math we did in school.

adding calculus, things like that, where a lot of it was about computation, calculation, applying things in a sort of rote way where you had these rules that you memorized and then you figured things out from there. And that is nothing like what research mathematicians do.

What research mathematicians are really doing is exploring mathematical truth in a really fundamental, abstract way. So they are trying to prove whether certain statements or hypotheses are true or false. And to do that involves a lot of exploration and experimentation. You develop this sort of intuition that coupled with

helps you build up these mathematical arguments. And that's completely different than just calculating things. You're exploring more abstract mathematical relationships, seeing how things connect in surprising ways, and...

Yeah, that involves a very different approach. We're talking about impossibly large numbers. We're talking about strange shapes, multiple dimensions. All of these things come into play in research mathematics. Yeah, exactly. Right. And you said proving statements true. And I think that's actually a point of connection with the way a lot of people experienced math during their educations was the idea of the proof.

And so this is, I think, the second kind of concept we should get our heads around before we start digging into this question about the impact of AI is when we talk about a proof, what do we mean? And I know in your story, you actually have a really great analogy for explaining what we mean when we say proof.

Yeah, so a proof is essentially a logical argument that mathematicians use to convince other mathematicians that something is true. And the analogy I used in the article is kind of like you're building up a cathedral from the ground up. So you start with the foundation. You're laying what's going to be the basis for everything else. That's your axioms in math.

They're these sort of basic assumptions that you're deciding from the get-go are true. So they're chosen quite carefully, and mathematicians agree that they're true. Right, okay. So something like Euclid, when he had his geometric axioms, one of them was like, between any two points, you could draw a line. That's the sort of thing. And there's no way to prove that. It's just, well, that's what we've decided is a fundamental truth. Modern mathematics, those axioms are a little bit more abstract. They're all in terms of things called sets.

But it's the same idea like that. It's almost like you're establishing the rules of the space that you're working in. And there's a very agreed upon set of axioms that mathematicians work with. So yes, that's the foundation of your building. And now you want to first create a blueprint for what you think you're going to do. So a sort of...

proof sketch or plan? What is the high level argument here? What's the arc of logic that I'm going to follow to build things up to the final statement? I think X is true. And here are the reasons that I think so. Exactly. But then you have to actually implement that blueprint. And so you need to brick by brick build up your building, your cathedral. And what those bricks are essentially in the language of proofs are these smaller statements called lemmas.

So a lemma might be something that mathematicians intuitively know is true, but they still need to show it very rigorously. And it's not axiomatic, right? It's not something everyone agrees is true, but it's something smaller that we feel like, okay, we need to make sure we establish this is true. Exactly. It's like a mini statement. So now we're going to create a little sub-proof of this thing. And okay, now we know that's true, so we're allowed to use this brick.

And you keep doing that with lots and lots of different bricks, and you have to put them together in clever ways. And eventually, you get your cathedral, and it stands on solid ground. And that's your proof. Now, you use the analogy of a cathedral rather than a warehouse for a reason, I think, because what we're talking about, and one of the things that this story is about, is beauty. The way that we think of modern math is not just in terms of

how do we get to this end outcome? But mathematicians, and I've learned this in being an editor at Quanta for a little while, mathematicians really care about the art of math, the beauty of math. Yeah, it's funny. Whenever I ask mathematicians, why is this problem interesting? Or why do you want to work on this for years of your life? One of the number one answers will be, well, it's beautiful. It's natural. There's kind of this understanding of

The proof is brief and elegant. You can understand it intuitively, even if it took a lot of hard work to actually get the proof. There's usually some really intriguing connection between seemingly separate, disconnected things or some other really fresh idea that is surprising to a mathematician. And that will be considered beautiful. How does this differ from applied math?

Pure math is about understanding math for its own sake. So when we say applied math, we want to use math for some practical purpose. So I have some engineering application. I want to make sure that my airplane will fly or that this building isn't going to crumble. Well, a lot of math goes into that, and that's really important to get right, but it ultimately has a different purpose.

Pure math research is different in that you don't really know what it's going to be used for. Maybe it'll be used for something practical 100 years later, the way that understanding the prime numbers has influenced cryptography today. Yeah. Maybe it won't be useful at all. But as we just talked about, it's still beautiful and worth exploring for its own sake. Yeah. The parallels between math and art, they just keep coming up like it's hard to escape them. Yeah.

A ton of mathematicians I talk to are also extremely interested in like poetry and literature. And that's always been a connection I've been fascinated by. Yeah. Let's take this landscape of pure math, creativity, beauty, elegance, but also the very defined structure of a proof, for example. And we start to think of now AI and what that has to do with it. It's notorious for actually not being good at math, but that's different than what we're talking about here, isn't it?

Yeah. So first of all, I think we also have to think about what is AI here. Okay. Because there are a lot of different types of AI systems that have been involved in math. If you actually look back decades, there are these systems called automatic theorem provers. Okay. And those are basically like you've programmed your axioms and these like rules of logic.

And these systems search over some big space to build up proofs of statements. And those work. They work nicely. And they've been around for a long time. But they're limited. The types of statements that you could prove...

They don't include some of the really big problems that mathematicians want to solve. Sure. And then you get these machine learning systems that could be used almost as a collaborator. So for the past few years, mathematicians have been using them to find new patterns or to test out possible conjectures. And that makes sense intuitively because throughout the sciences, some of the best places that AI has found great application is when you have a gigantic data set and you're looking for patterns within it.

Like, it's easy enough to say what's in here and have an AI do some of that work for you. And that's exactly what those systems are doing. Yeah. Now, what a lot of research mathematicians, as we spoke about, are spending their time doing is creatively thinking about things in a – maybe a less logical way at first, like before it becomes a proof. Right. And so then the question is, can AI help with that process? Like, can you actually –

tell it, hey, I want to prove this statement, give me a proof. And we're not there yet, but there are some nice benchmarks that these systems have passed. Which is putting together the rigor of the automated proving systems from before with that sort of generative, quote unquote, creativity that we see out of

an LLM or an image creation model or something. Yeah. And the question is, okay, with an LLM, which is essentially trained to complete sentences almost, can something like that, which isn't actively thinking or being creative in the way that we think mathematicians are being, can it prove the kinds of things that mathematicians are doing in their very human way? Yeah.

And so that's a question going forward. - And just to revisit something from earlier, the reputation for LLMs in particular to be bad at math, because that's the interface that a lot of people have with our latest AI systems is through LLMs. And if you start asking them questions involving big numbers,

they tend to get more and more unreliable. Yeah, they are very unreliable at even basic arithmetic with very large numbers. They also are unreliable if you're asking for a mathematical explanation or even like a proof of a known statement where the proofs exist. So it's been trained on those. It can maybe like give you the elementary proofs, but then if you tweak things slightly, it'll sound right when it gives you what its answer is, but it's really wrong. And so if you don't quite...

know how to query it, you'll get the wrong answer. And so, yeah, that's where it's at. I think that there are more bespoke models that are better at these sorts of things. And so I think some mathematicians are actually kind of impressed at where things are. And last year, some of this LLM technology was used in conjunction with other systems to do really well on the International Mathematics Olympiad, which is this very prestigious high school proof-based exam.

But again, those are all very different types of problems than what mathematicians are dealing with in their everyday life. Now, I know that as a writer, when these LLMs first emerged, everyone was very concerned about

How do we relate to these things? Are they tools for us? Are they coming for us? And that's faded a little bit because I think the quality maybe isn't quite there yet. I think artists and illustrators have experienced something similar. And it sounds like from your reporting on this story that mathematicians are going through a process of their own that says –

If we're no longer solely enmeshed in thought, trying to devise these abstract questions and then build these cathedrals of proof for them, then what are we doing here? Yeah, I remember one mathematician describing it to me as like, oh, it makes you feel a little queasy because we like to just sit in a room with nothing around us and think deeply. And that's not what this would look like if AI were to encroach on how we're doing stuff. But I would say probably most mathematicians are kind of indifferent right now because we're

These systems are not quite good enough to really be making a huge impact in actually proving things. But then other mathematicians are exploring, well, how could it already start to help us and what might that future look like? What do the optimistic mathematicians think the future could hold?

I think there's some hope that, okay, maybe some of those rote lemmas, these bricks that make up the cathedral, can be outsourced the way we currently outsource a piece of arithmetic to a calculator or a computer. This seems like a natural extension of where technology takes us. It's things that...

We could do, but we're really super time consuming. And here's a more advanced thing that would save us some time. Yeah, exactly. So that would free up time to explore new things. And so mathematics will happen faster. But it will also mean that there's this step away from being involved in every rigorous step of the proof. And that's been a very defining thing for mathematics for decades.

much of the 20th and 21st century. There's a sense in which getting up to your elbows in this kind of work actually could lead to greater insight because you're having to think your way through things on a regular basis. What about the pessimists in this category? Some of the mathematicians I spoke to, and this didn't actually make it much into my piece, but they're concerned about how because

Right. Yeah.

And then there's this question of what if the AI tech just gets so good that it can actually prove things? Like we had these problems that we were really interested in and devoting our lives to. And if you could figure out a way to feed it to the computer correctly, the computer can output it.

And this is a total fantasy right now. It's unclear if this will even be possible. There might be just completely insurmountable barriers on the tech side. But it's within the realm of imagination. Yeah, and mathematicians are willing to think about it as they should. And so then there's this question of what will we be left to do? If we're spending all of our time proving theorems and that's where we see our creativity at work, what will be left?

And there's no guaranteeing that the proof that comes out of that is going to be beautiful or elegant or efficient or any of the things that mathematicians seem to prize in proofs. Right. And then there's a question, well, will what mathematicians consider beautiful, will that change? How will that adapt? Or will they have to find beauty in other aspects of the enterprise? It sounds like mathematicians are starting to think through, OK, maybe there's just a different model for us than the one that we've worked with for so long.

Yeah. So the collaboration idea is definitely a big one. Right now, the kind of norm in math is that you get two or three mathematicians and they work together from beginning to end on a proof. Okay. So they brainstorm, they come up with the ideas, they figure out how to implement it. And so they together create this entire proof and figure out how to write it and communicate it to other mathematicians. And...

Now there's this question, well, if you could outsource a lot of this stuff to AI systems, this could open up these really, really huge projects instead. We've actually, we've seen this before in math, even without AI. So the like classic example is this huge project in the 20th century called the classification of finite simple groups. Okay. It's very fundamental objects in math that mathematicians wanted to classify. That took...

Basically, over the course of that whole century, like 100 mathematicians all working separately were able to accumulate what ended up being the proof. And it was like more than 10,000 pages. Wow. So no one person now holds all that in their head, but everyone just considers it true and is comfortable with that. That sounds like good proof.

for a version in which you're doing parts of a project, AI is doing parts of a project, you're working with other people, like thinking in that grand collaborative way. Yeah, exactly. And it might become the norm where you have tons of people kind of experimenting and testing things out. And then some people are solving one part, some people are solving another, other people are figuring out how will I feed this properly to the AI system and interpret the results. And so you get...

This division of labor that's quite common in like physics and biology, but right now it's not common in math. I think this is going to be a place we're going to continue to explore in pieces over the years. So I'm excited to keep talking about this with you in the future, Jordana. Yeah, me too. I mean, everything's moving so fast, no one really knows what's going to happen next. And so it's an exciting time. ♪

We like to close every episode with a recommendation. This can be some piece of media out there you've consumed, something you think is really interesting. What's captured your imagination this week? Yeah, so to keep the AI theme going, I read an article in New York Magazine called...

Everyone is Cheating Their Way Through College, which is a very provocative title by James Walsh. And it's about how students are using LLMs to write essays and complete other assignments in college and how that's affecting what higher education might look like and how people are actually learning and thinking now with these technologies entering their lives much earlier. That's very interesting. Thanks so much for joining us, Jordana. Looking forward to the next time we get to talk.

Thanks so much. We don't just report on AI at Quanta. Not all of our stories are about that. So if you do check out quantamagazine.org this week, you might also find stories about turbulence in fluids, how birds migrate long distances, and plenty of other amazing stuff. So check it out on site and check out Jordana's story when you get a chance.

We're going to leave you today with a sound that rings out over the final resting places of some of history's greatest scientists, Charles Darwin, Stephen Hawking, and Isaac Newton. These are the bells of Westminster Abbey. The Quanta Podcast is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. I'm Quanta's Editor-in-Chief, Samir Patel.

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 Quanta podcast is produced in partnership with PRX Productions. The production team is Ali Budner, Deborah J. Balthazar, Genevieve Sponsler, and Tommy Bazarian. The executive producer of PRX Productions is Jocelyn Gonzalez.

From Quanta Magazine, Simon France and myself provide editorial guidance with support from Matt Karlstrom, Samuel Velasco, Simone Barr, and Michael Kenyongolo. Our theme music is from APM Music. If you have any questions or comments for us, please email us at quanta at simonsfoundation.org. Thanks for listening.