Terence Tao Talks to Numberphile About AI

**TL;DR: Terence Tao believes mathematics is moving from an era of "proof scarcity" to an industrial age of "proof abundance." The problem is that human understanding is being sidelined, while the incentives of AI companies and academia are becoming dangerously misaligned.** ## What happened in early October? **Host:** What do we call this big thing that happened in early October? OpenAI released hundreds of proofs. Does this event have a name? Usually, an event this significant gets a nickname or a name. **Terence Tao:** I think it will eventually have a name. We're still digesting it. There isn't a standard name yet. What I've been saying all year is that we're transitioning from an **era of proof scarcity** to an **era of proof abundance**. For thousands of years, proofs have been hard to come by. The people who found them needed understanding, expertise, a bit of luck, communication with others, and familiarity with the literature. So proofs have always been widely valued—everyone who proved a difficult theorem put in hard work, and that helped move the whole field forward. And now we've found a shortcut. We've found machines that can obtain these proofs for us, and we're now drowning in proofs without gaining the understanding that used to come with them. It's like going from an era of food scarcity—when we didn't have much food, but every bite was nutritious and healthy—to being inundated with synthetic food that is technically edible but isn't as satisfying as what we had before. ## The end of mathematics' "heroic age" **Host:** Do you feel lucky to have lived in the era of proof scarcity and had the opportunity to produce some major proofs? Do you feel like you got in just before that dividing line? **Terence Tao:** Yes. I think we're witnessing the end of the romantic age of mathematics, the end of the heroic age. This happened in other disciplines centuries ago, going back as far as the Industrial Revolution. But somehow mathematics held out, and we have some unique traditions. For example, we all use blackboards—you can see I'm using this special Japanese/Korean chalk. We have these quirks, like spending enormous amounts of time mentoring graduate students one-on-one. We have these almost mythical stories, like Andrew Wiles spending seven years in an attic. Your YouTube channel has covered many stories like that. I think we're entering an industrial age in which mathematics will operate on a much larger scale, and we'll work in bigger teams, more like other scientific disciplines—which have already had to adapt to industrialization. Some things will be lost. There really was something romantic about that heroic age, and hopefully we can preserve some of it. At least in our education system, we should emphasize traditional mathematical methods first and then introduce the industrialized approaches, just as we teach children to do arithmetic by hand before giving them calculators. I think we need something similar. But yes, it is quite poignant. That era was wonderful, though slow and perhaps inefficient in some respects. And hero worship has its drawbacks too; there are winners and losers. But in any case, we're entering—rather reluctantly—a new era in which exciting new things become possible, while some things are being lost. The question is what valuable things we can preserve and what we can let go of. ## Two paths forward: understanding and technology **Host:** You've been using AI for a while. I've seen you talk about AI in lectures. The attitude I've picked up from your talks has been quite positive. Did you foresee all of this? **Terence Tao:** I saw two paths forward. Mathematical progress has two dimensions. One dimension is **understanding**—whether humans understand what is happening. The other is how much **technology** you use—whether you're using pen and paper or all these technologies. In the past, we didn't have technology that could solve problems for us. We had things like calculators. We could solve some mathematical problems, but for most mathematical problems, we needed pen and paper. So we progressed by developing understanding. We advanced along the understanding dimension. Around 2023, I began seeing ChatGPT emerge, along with some other developments. There was a formal language called **Lean** that was very exciting. I could see that this technology could really make many of our mathematical tasks easier and more efficient. But the question was whether we would **adopt these technologies while maintaining our understanding**, or **sacrifice understanding in order to use these tools**. It's like inventing the car: do you start driving and stop exercising, or do you find a work-life balance—sometimes driving, sometimes walking? I could see one path being very unhealthy, but another being very exciting. So I've been encouraging people to embrace these technologies and explore how to use them in healthy ways. But I think events have overtaken us. Enough people and companies are single-mindedly pursuing the speed, output, and performance of these tools, putting everything else second. They're willing to cast understanding aside to prove more theorems, issue more press releases, and test their latest toys. That's dominating the news now. Hopefully we can return to a more balanced perspective, but at the moment, "raw AI, nobody understands anything, just keep accelerating" is the prevailing mindset. I don't think that's sustainable. At some point, the lack of understanding will have real consequences. In mathematics, perhaps it's not so bad yet—if one of our favorite problems is solved by AI rather than humans, and we don't understand it, many mathematicians will be upset, but the rest of the world can cope. For example, with the Navier–Stokes equations, if we have some "blob" of a solution that nobody really understands, that's okay. But if we start using these AIs to model actual weather or power grids, and they keep giving us recommendations we don't understand, then at some point that will cause real problems. ## Who should the beacon problems be left for? **Host:** One argument I often hear from the mathematics community is that these special problems, these conjectures, these problems often described as beacons or milestones, are very important to the field as a whole and should be left alone, untouched by AI companies. Is that a position you've expressed or advocated? **Terence Tao:** Yes. To some extent, this has to do with how we train the next generation. We've built this system—a very standard way for students to learn mathematics and earn professional recognition is to solve a moderately difficult problem, perhaps one that someone more senior could solve themselves, but there's an unspoken understanding that we leave certain problems for students. They're not explicitly assigned, but there's a consensus that certain classes of problems are appropriate for students. If you're not seeking professional recognition, you work on other problems. So there are certain types of training problems that should be left for students. This wasn't a problem in the past because there were so many problems, relatively few students, and the problems were difficult to solve, so there was no shortage. It's like in the days of sustainable fishing: you didn't need to allocate particular fishing grounds to particular people. There were enough fish in the sea for everyone. But now we have this kind of **mathematical strip mining**—these enormous machines that can blast away hundreds or thousands of problems at once, like fishing with explosives. Suddenly, we don't have good problems to assign to people. People used to share lists of problems as challenges for others, to motivate students. Paul Erdős, for example, was famous for this. But now people are reluctant to share their favorite problems because someone with an AI prompt will swoop in and solve them before the original purpose can be fulfilled—namely, training a student or postdoc and helping them publish their first paper. The incentives of these AI companies and random individuals using these AI tools are no longer aligned with the traditional incentives of our mathematics community. At least one side has to change. So internally, I think we have to change our own incentives. I think we need to move toward more collaborative projects, where instead of each person working on a single problem, a team works on hundreds of problems, so it matters less if someone gets scooped on a particular one. But I also think the broader community needs to respect some of our spaces. It's like organizing a marathon: people generally don't drive along the same road at the same time and overtake the runners. In principle, you could do that, but everyone would consider it socially inappropriate. ## Analogy as the tool of choice **Host:** Terence, you're so good at analogies. Do you come up with them on the spot, or have you had a lot of practice from doing so many interviews? **Terence Tao:** A bit of both. But my entire mathematical style is actually based on analogy. I think Isaiah Berlin once drew a distinction between hedgehogs and foxes—some scholars are foxes; they know a little about everything. Others are hedgehogs; they do one thing extremely well. I've always been a fox. I like taking ideas from one area of mathematics—say, ergodic theory—and applying them to another, such as combinatorics. The language is different, the notation is different, but if you're good at analogy, you can understand one field through another, and that works very well for me. I like analogies; they're my tool of choice. ## The toll of the past few weeks **Host:** Starting with Navier–Stokes and then this explosion of proofs, so much of it seems to have revolved around you—even though you weren't involved in producing any of it. You've published a lot of guest posts on your blog. What toll have these past few weeks taken on you? **Terence Tao:** It's been very surreal, very crazy. Every hour of every day is booked. I'm still teaching and talking with students, which is actually the most normal part of my day. Then there are interview requests. I write articles for the media. I give talks and make lots of slides. I've done almost no mathematical research except through my students and postdocs—they're still doing traditional mathematics, and I'm advising them, but they no longer have time to think about actual mathematics. I'm in all the group chats. One strange thing is that I'm suddenly talking constantly with hundreds of mathematicians. I'm in a group chat with 20 Fields Medalists. That's actually a very interesting group chat. We should have set it up years ago. **Host:** I'd love to be in that group chat, but I'm sure I wouldn't understand what you were talking about. **Terence Tao:** Yes. We should have more group chats like that. We show far too little of these behind-the-scenes stories—how the sausage is made. I think this is actually something we'll see more of in the future, because these AIs can generate the finished products, these beautiful papers, but they're very bad at showing the process of how you got there. To be fair, humans aren't good at that either, at least historically. We also produce these beautiful papers without showing much of the process. We don't invite documentary crews in. Occasionally there are open projects, like the Polymath Project, which you may have heard of. But I think we'll actually see more projects like that, because that's something more uniquely human. ## Are AI companies the enemy? **Host:** Are AI companies the enemy? I know you've worked with them and had good relationships with them, but now we're hearing a lot of negative things about them—chasing press releases, chasing company valuations, and potentially appropriating research for their products. **Terence Tao:** First of all, I know some individuals at these companies. Many of them actually come from the mathematics community. I consider many of them friends. Many have very good, balanced views on how AI should be used in science. As I said, for example, using AI to **enhance** human understanding rather than replace it. There really are many individuals with that mindset. But I think what we're realizing is that the cultures and incentives of these companies, especially the largest ones, differ from those of academia. For a while, that was fine. It's okay for people to have different goals and incentives, as long as all those goals are still far off. You can move in the same direction, advancing both their goals and ours. That was the case until around 2025. When these technology companies were able to use these tools to do some mathematics, they did so in collaboration with mathematicians because the tools weren't yet powerful enough to do it independently, and it was basically a win-win. Everyone was more or less happy. But now the tools are so good, so effective at optimizing, that we're seeing this **misalignment**: if you keep optimizing for what technology companies want, they're diverging from what the community wants—things like truly understanding mathematics, communicating with the next generation, and sustaining a human community capable of posing new questions. These AIs are consuming our pool of problems without replenishing it with anything new. In the past, that happened naturally: someone who solved a difficult problem would naturally come up with two new questions as follow-ups. Source: Terence Tao Talks to Numberphile About AI (https://www.youtube.com/watch?v=bAnwMdfcMVw)