Hi everyone and welcome back to the FAQ podcast, where we have casual conversations with extraordinary people across science and technology. I'm your host Tyna Bradley, and today's episode is about a pretty hot topic right now, and that is AI and mathematics. Now, if you were to spend any amount of time on X or Reddit or just searching the web for the latest news in AI, then you'll definitely know that this is a thing right now, and there's a lot happening very fast. If you're in this field, then you'll probably know exactly what I'm talking about, but in this episode I wanted to help get everyone else up to speed just by having a casual conversation with a mathematician who's been actively using AI — by which I really mean large language models, basically — since the beginning. And that mathematician is today's guest, Daniel Litt, an assistant professor of mathematics at the University of Toronto.
Now, I didn't ask Daniel to speculate about some of the trending topics because there's already a lot of that happening in the public discourse right now. Instead, I really just wanted to chat with him and hear how he's thinking about things, especially as someone who uses AI in his research. What does that look like? I wanted to hear about his workflow, a day in the life kind of thing. I also wanted to hear about best practices that are currently useful for him and possibly for others, and just generally how he's thinking about the mathematical profession today. As someone who works in math, I honestly left this conversation feeling motivated to go try out some new things, but no matter your line of work, I do hope you'll enjoy this peek into the life of one mathematician in the age of AI. Let's get to it.
So, um, I think, Daniel, a question I wanted to open with — and I'm sure that you saw this — but there is a Nature interview with Terry Tao recently, I think like in April of this year, um, and one of the questions was basically like, what are mathematicians' attitudes towards AI? Are they, are they embracing it these days? And, um, you might remember Terry Tao's, like, you see all five stages of grief play out. There's like denial, and then anger, and then bargaining, and then depression, and then acceptance. And I thought that was kind of funny, but then it made me curious. What, like, would you say that's been your experience? I think, you know, we know now how you're, you're kind of thinking of AI. I think it's like beyond acceptance. You're like really embracing it and kind of at the forefront. But just thinking back over the years, would this like describe you, or what has your experience been like?
Yeah, I mean, I think that's probably an accurate description of some mathematicians' reactions to the existence of new AI tools, and I don't, I didn't find it particularly relatable. Like, I think I was initially skeptical of the tools' usefulness, and of course now they've become much more capable and now they're useful. And, you know, as you said, I do genuinely use them for my own research. But yeah, I don't, I didn't feel like I had a huge emotional journey as part of that.
You know, I think probably one's reaction to the existence of highly capable tools for math research depends a lot on one's personal relationship to mathematics. In my case, I kind of think of it as like a job where the goal is to do good science, and, you know, a year ago the tools were not really at the point where they could help with that beyond, you know, some sort of very small tasks like literature search, like along the lines of using Google or something like that, and now they are. So, you know, the change has been that, like, well, the tools become more useful, and so I'm using them more.
I think there are people who are like concerned about changes to the structure of the profession, and I am concerned about that, but I don't think that's like, uh, you know, grief is not the accurate description. Like, I think there are a lot of ways academia and math academia are structured imperfectly already. Um, they're certainly not, you know, currently resilient to massive disruptions. Like, I think we should expect it to come from, you know, increasingly capable AI. Uh, but okay, that's also an opportunity to fix some of the problems.
Yeah, that's good. And I do want to ask you maybe later your thoughts more on that about, you know, how the mathematical profession is changing. But, um, I want to maybe continue to think backwards in time. Can you remind me, when did you start experimenting with AI and LLMs? Was it right when ChatGPT came out? Was it before that?
Yeah, so it was really at the beginning. So I think initially ChatGPT 3, when it came out, it was not publicly available except through a kind of computer game called AI Dungeon. The idea was that ChatGPT would like be a dungeon master for a sort of Dungeons and Dragons game. And so I was curious, and I bought that, and I like started trying to get it to do math like in the middle of a Dungeons and Dragons adventure. [clears throat] So that was my first experience. So I think that must have been like 2022.
And yeah, now, you know, four years later, it's very different, right? Like at the time, I think the models couldn't, you know, like add or multiply. Like so you would ask it, you know, I would get in some situation where my character was a math professor, and then I would say, 'And then he added two and two.' And the model would reply, 'Unfortunately, his calculator broke, so he was unable to perform that addition.' [laughter]
You know, and now they can be useful for like algebraic geometry.
Okay, that's so funny. I didn't realize you were hijacking the game that way, but that's awesome. Okay, good. Well, I want to — I was asking you about like thinking in the past. Okay, let's look to the future. There was a symposium at Stanford recently called the Future of Math or Future of Mathematics. I'm assuming you tuned into that, or maybe you were there in person. Did you...
I unfortunately couldn't make it, but yeah, I'm aware of it. I didn't watch the whole thing, but I've seen parts.
Yeah. I think so Terry Tao gave a talk there which I watched, and probably you've heard people make reference to this, but I think I'm paraphrasing now. I think he said like, you know, one problem mathematicians might be facing in this new era is like proof indigestion. It's not like we have to digest someone else's work like when you're reviewing a paper or chatting with a colleague, but like we have so much information now because AI can just crank out stuff and it's not always correct. In fact, mostly it's probably not correct. And so there's a little bit of indigestion there. I think you made reference to that once on X, which I really liked. I think you said something to the effect of like it's hard to understand what other people have done, right? It takes like a lot of work.
Yeah. The dirty little secret of mathematics is that like no one understands anything anyone else is saying.
You know, you write some like, I don't know, I recently wrote like a 93-page paper. But I understand it, my co-author understands it, but you know, I think it's very hard to communicate. Like there are a lot of mathematical ideas in that paper, and like asking someone to digest them, it's like a huge time investment. So I think, you know, people will try to do that. Like, you know, the paper will have referees, and then, you know, as it penetrates the psyche of the community, like people will engage with it in various ways. I think very little of that is like reading a paper from start to finish.
So, like a lot of the way you engage with a paper is you kind of imagine you read it. Like, you try to re-prove the results yourself or understand what the big ideas are, and you use the text of the paper as hints, but a lot of stuff is happening internally, right? So now, you know, we're starting to see AIs generate — well, first of all, actually correct proofs of some conjectures, and also, as you say, a lot of incorrect stuff, but, you know, more and more correct things.
Or, you know, in my own work, like I'll discuss some topic with the AI, and maybe it will generate a proof of whatever or something, which may or may not be correct. And so, yeah, there's somehow a new — I think what Tao calls it is an era of proof abundance, where we now have tons of proofs, many of which are correct and many of which are not correct. And yeah, so this interacts very poorly with the fact that it's impossible to understand what anyone else has written, right? Because there's now a huge new amount of text that we have to digest. You know, I think there's a question of like how to adapt to that.
And part of my view is like, well, what is the purpose of proof to begin with? It's like to ground our mathematical understanding in kind of truth, right? Like, it's very easy to just go out and start bullshitting if you don't start proving things. But like the point of math is not necessarily just to prove things, it's to understand things, and I think we can ground our understanding in ways other than just purely like checking a proof in detail.
So, for example, one new sort of benefit of the use of the AI tools is that it suddenly became much easier to code. So like, you know, one thing you should do when you come up with a new result is you should prove it, right? But like another thing you should do is like work out a bunch of examples. And historically, in some areas like mine, that latter step was often much harder. Like just getting your hands on some kind of complicated mathematical object in a way that's like, you know, beyond the first sort of baby cases, like where you can really understand an example, is challenging.
And if you wanted to write a computer program to do it, it would maybe have taken me, you know, many months. But now, you know, I have all these projects where because of like live coding tools, just coding in general has become much easier and faster. Like, you know, I'm much more willing to put in, you know, a couple weeks or a month of time to code up a sort of fairly involved examples. You know, so I think we can start thinking about ways to verify and kind of get evidence that a statement is correct that go beyond like, you know, informal mathematical proof.
Yeah. Actually, so I'm glad that you mentioned that, like the idea of spending weeks at a time to do something like this. Can you actually just paint a picture of like a typical day for you as a mathematician who's using AI? Like what — you're in your office, you're sitting at your desk, you have books, maybe there's a computer in front of you. Like what's going through your head as you're thinking about your work and using Codex or GPT or whatever? How does that look like for you?
Yeah, so first of all, I do want to stress, I think, you know, these tools are pretty useful, but it's not like I'm using AI all the time. Like pen and paper is still actually, for me at least, the primary way I understand what's going on. Yeah, so maybe let me say first of all, what does like at least the average day for me look like? It is like a lot of staring up into space.
Like from the outside, I don't know, maybe it looks like not a lot is going on. And sometimes even from the inside it feels like, you know, not a lot is going on. My head empty.
But yeah, I mean, okay, so typically, at least like let's say near the beginning of a project, you know, there's some kind of question I'm trying to understand. I have some idea of how to move forward. And so like there's some literature search aspect maybe where I'm trying to understand what's known. Actually, AI tools have been useful for quite some time. So, you know, I wrote a paper that went up on arXiv in January where one of the main ingredients was some hyperkähler geometry. Okay, it doesn't really matter what that is. I didn't know any hyperkähler geometry before starting this project. I just had some sense it would be useful. And so I learned it over last summer with much help from o3.
So there is something where — so ChatGPT o3 was one of the earlier reasoning models. So there is something where it's like more or less substituting for Google and reading some papers. It made lots of mistakes, and so, you know, that's something you have to be on your toes about. But that was already quite useful just for sort of learning and literature search tasks. So that's sort of one early way the sort of new AI tools started to become useful.
Yeah, so the other primary way the tools are useful, like when I'm not just like scribbling on a piece of paper or just thinking about something and staring into space, is like — yeah, so, you know, when you are trying to understand some mathematical phenomenon or come up with a proof, like typically you kind of have some overarching maybe proof structure in mind, or you kind of are testing some ideas as to how something might work. Right now the models are very, very good at finding counterexamples to possible statements. So one thing I have started doing is asking — you know, if I think something might be true, I would just ask the model, 'Could this be true?' And, you know, sometimes it will find a counterexample, that's very useful. Similarly, just finding examples is sometimes kind of hard.
But just as an example, my student Charlie Wu, his thesis is about classifying compact components of relative character varieties, whatever that means. And so he came up with some beautiful classification in terms of some objects he called crest variations, and we found a bunch of crest variations by hand. It was kind of like an artisanal process. So like, you know, it would take us, you know, like many, many hours or days of computation to find one example.
And so this was maybe like six months ago, and then over winter break — I think this is sort of when everyone discovered coding agents — I decided I would try to automate the process a little bit. And so I kind of coded up — I vibe coded a tool using Codex to compute with parabolic Higgs bundles, which are some fancy object in algebraic geometry. This is like a scientific computing task I would not have even considered trying a year ago. Like it just would have been very, very involved. And so, okay, with the new tools it took me like a week.
Of course the tools made lots and lots of mistakes, and so, you know, it was sort of challenging to adapt the workflow to that, but okay, eventually I got something that worked. Okay, and we still had trouble finding examples actually, so somehow this was quite challenging. And then, okay, I've been working on these tools for the last, I don't know, four or five months, and then now finally we've started to find new sources of examples, and that's been really, really useful. I think it really improved the — like we much better understand the structure of these objects right now.
Yeah, so but I think the other main way the tools are useful is just like rubber ducking, kind of. Like, my favorite way to do math is to have a collaborator and talk to them about it. But, you know, a human collaborator often gets, you know, tired or, you know, has other stuff on their mind, and so it's just very useful to have a smart, you know, mathematically capable entity to talk to.
My sense is that the models are overall quite uneven in math. So like, you know, in some areas of combinatorics, for example, or geometry, discrete geometry, they're able to autonomously do like relatively interesting mathematics, as I understand it. In my area, it seems like thus far they're a little less capable, though, you know, who knows where things will be in six months. But yeah, they're still useful to talk to, and, you know, they can, you know, rule out stupid ideas or point me to some literature I wasn't aware of, and that can be quite — you know, that makes a big difference.
And do you primarily like GPT? Do you use Gemini or anything else? Do you have a favorite?
I've tried all the frontier models, so Claude Opus, ChatGPT 5.5 Pro, and Gemini Deep Think. I have gotten most use out of ChatGPT 5.5 Pro, which seems to be slightly better for most mathematical tasks.
I think partly that's like a function of reliability. Like in the last few model iterations, like maybe since 5.2 Pro or so, the models make a lot fewer mistakes. And that's like a huge — you know, even if they're not much smarter, although I think they are also getting smarter, that's just a huge boon to reliability.
That said, so I recently wrote a paper in which somehow the AI contribution was relatively minimal, but like we proved the main theorem without using AI tools at all, but then there were various sort of natural questions that arose after the fact that we used AI to answer. And it actually turned out, for whatever reason, Gemini DeepMind and Alethia, which is an internal scaffold at Google developed by Tony Feng, turned out to be much more useful than ChatGPT 5.5 Pro. So we had sort of various questions that came up in the course of research, and those ended up being answered by models from DeepMind and not OpenAI.
Well, I mean, I think those aren't publicly available. Like, if any mathematician listening was like, 'Oh, let me try that,' I think we can't yet, but maybe...
Alethia is not available, but DeepMind is, although it's $200 a month. Like, same with ChatGPT 5.5 Pro.
Yeah, I mean, one thing to note is that the free or cheap models are like almost useless for math. Like, I think unfortunately it seems to be the very expensive models so far that are more useful.
Oh, okay. So that's good to know. Let me also ask you along those lines. Okay, let's imagine mathematicians that are maybe listening and are inspired. Okay, so Daniel like tinkered around for a long time, and then the more you kind of play around with these things, the more you see what they're good at, and you start to, I think, probably develop like a list of personal best practices. So, for example, you said like you can just ask the model like, 'Hey, is what you just wrote correct?' And then it can like double-check. And maybe that was not so reliable in previous models, but maybe for a 5.5 Pro, like if you ask it, 'Hey, double-check your work,' it actually will find a mistake or something.
Can you — okay, good. Are there other best practices that you've kind of like got in your back pocket that maybe you find is helpful? Let me give you another one. I heard you say this, I think maybe in a seminar talk you gave recently, but you're like, 'Look, it's hard to check if a proof is correct that an LLM gives you, right?' Because you have to sift through all of this stuff that looks plausible, but when you actually think it's wrong — and maybe that's frustrating because it's a waste of time and you just spend hours like reading ChatGPT's proof and it's wrong. So I think at that time, and maybe this was last month or something, you're like, 'Hey, if you see something you don't understand, just stop.'
Not worth it. Just prompt it again and then see what you get.
Maybe that's a best practice. Do you still stand by that? Are there other things that you think would be helpful for people?
Yeah, so I would say that you've learned — that's a good question. Yeah, so one comment is that, yeah, the models are changing very rapidly. So what might be best practice one day may no longer be best practice. So there's definitely some learning curve. That said, I think because things are changing so rapidly, if you just start experimenting, you're not really behind someone who's been playing around with the models for months.
Yeah, so in terms of that specific comment, like yeah, so I think what I suggested is like — right, so you're trying to check the output of an LLM. Right, so yeah, first of all, ask it to check itself. That already bumps up reliability hugely. And then the other thing I suggested at the time was that like when you, you know, if it says something you don't understand, it's in an area of your expertise, it's probably just wrong. Like just, yeah, prompt it again.
I would say that turns out to seem less true with the latest model generation. So particularly 5.5 Pro seems to be a bit more accurate. It definitely happens sometimes that it'll, you know, say something that doesn't make sense and then you waste a lot of time. I think it's, yeah, I would say I stand by that remark, but on the other hand, you know, if you had it generate output 10 times and you have it check itself and it seems confident, then it might actually — now to the point where it might actually be worth engaging with and trying to understand what it's saying. Okay, it might actually be correct.
There's sort of a weird paradox here, I think, which is that like when the models were very unreliable, it was very easy to dismiss sort of difficult to understand model output, and like that would typically save you a lot of time. As the models become a bit more reliable, now it's sometimes worthwhile to engage, but on the other hand, that means when they're wrong, you waste a lot more time.
So, for example, I recently, you know, I said could such and such be true to ChatGPT 5.5 Pro, and it gave me a pretty classic counterexample which came from some paper of Kondo, which I wasn't familiar with. And so I was like, okay, well, that could be true, I guess. Let me look at this paper of Kondo. Okay, so I had to read this whole paper, which is like a very substantial endeavor. It wasn't that long a paper, but that, you know, takes a couple of days to really engage deeply enough to figure out what's going on. And then it turned out ChatGPT 5.5 Pro was wrong. So like it wasn't actually a counterexample. So, okay, like that was a huge side investment of my time. [laughter] It wasn't exactly wasted time because like I read a cool paper, but yeah, so somehow as they become more reliable, like sometimes you do end up wasting more time actually when they are wrong.
I think, yeah, I think the thing that takes a while to get used to is first of all, like there's now a huge abundance of text. So like there's nothing wrong with throwing away model output. Like just throw it away. If like you think you can get it to generate something better, like just throw it away and start over. Like that's something like, you know, you couldn't have done with a person's paper before. Like someone writes something confusing, like you can't just throw away the paper and get a new paper. Like you have to figure out what's going on. Like it's an email with errors. But that's a huge amount of their time.
But yeah, now, you know, the models can do the same thing 10 times in parallel, then you can give those 10 outputs back to the model and ask it to judge which is correct and so on and so forth. Like that's something you would never have done before, and it's worth kind of getting used to that.
I think the other maybe comment I have along those lines is like, I think one challenge I had when I was starting to use the models, and I think one challenge many mathematicians have, is like the models are now starting to get quite smart, like genuinely useful. But also they — like the extent to which they're truth-seeking is limited, although maybe improving. And like if you were to interact in a professional capacity with someone like that, you would be like, I'm never going to talk to such a person again. Like it's kind of like it's just, you know, it's not pleasant. And I think in math especially there's like a culture of like not putting up with that.
And yeah, so I think somehow overcoming that is one of the big challenges with working with the models. And I, you know, part of the way one does that is just like trying to minimize the amount of BS one ends up actually reading by asking the models to check themselves or, you know, throwing away suspicious output and that sort of thing.
Right, right. You know what this reminds me of a little bit? Do you know the website called the nLab?
Yeah, okay. So do you remember like years ago someone made a pair — okay, actually, hold on. For people who don't know, so the nLab is like this Wikipedia-style website where you can look up math, you know, math concepts in category theory specifically. And it's kind of funny because you can look up like a relatively simple math idea,
But the way that is described using these very high level of abstraction, it's kind of comical how complicated.
Do you know about the NLabification functor?
The NLabification functor? I don't know. Is that related to the...
It's the left adjoint of the inclusion of obscure mathematics into all mathematics. [laughter]
Okay, I should have guessed.
