Transcript [00:00] - All right, so this is a kind of emergency episode [00:04] I think it's fair to say, about the state of mathematics and artificial [00:07] intelligence. A kind of crisis moment [00:11] in the history of mathematics. Over the last few weeks, [00:15] machines claimed to have solved some of the hardest problems in the [00:19] history of math. Navier-Stokes, the quasi- Riemann [00:23] hypothesis, the unique games conjecture, and hundreds of other results. So [00:29] here with me, we have Steven Brunton, [00:33] who's a mathematician, fluid dynamics expert at University of Washington, [00:38] who can speak to Navier-Stokes, turbulence, all things fluid, [00:42] and many other topics. Also, here's Ankur [00:46] Moitra, a mathematician and theoretical computer scientist at MIT, [00:50] visiting Institute for Advanced Studies in Princeton this year, [00:54] who can speak to many things, unique games, [00:58] conjecture, and complexity theory in general, [01:01] and the mathematics of how the AI learns and [01:05] all things theoretical computer science. And Alex, a [01:09] mathematician, number theorist at Rutgers, who can speak to [01:13] also very many topics, including Riemann hypothesis [01:17] and Lean, the automated proof and the automated [01:20] verifier and checking and so on, and so many other topics in mathematics. [01:25] All three are brilliant. All are very kind to show up this early [01:29] morning and talk to me, assemble this kinda [01:33] multi-agent mini swarm of human brains to try to interpret what happened [01:37] in this very moment. A lot of mathematicians, brilliant mathematicians, [01:41] including Terence Tao and others, are freaking out. It's a mix of awe, [01:48] terror, dread, and few inklings of light, of hope also. So I apologize to start [01:58] with a question about feeling, but what have you been [02:02] feeling about this historic moment? Maybe Alex, do you wanna go first? [02:08] - Yeah, I don't know if I would call it a crisis. There's a lot of opportunity in that crisis. [02:12] It's a chance to rethink what it is we're trying to accomplish, [02:17] and it's the chaos of opportunity that some [02:21] people are reacting negatively to, some people are reacting [02:24] positively to, and sometimes those reactions are happening in the same person. [02:29] And so I think we're entering, if we're not already in, a [02:33] period of some kind of golden age of mathematical knowledge [02:37] after millennia of having a [02:40] candle that we are walking around the woods, and this is as far as we can see. [02:44] Now we have a flashlight. [02:46] - I think we should comment on the magnitude of this moment for people who are not aware. [02:50] So maybe to quote you, something like something [02:54] like, "Holy shit, if a human did this, it'd be an instant Fields Medal." Maybe you didn't say, [02:58] "Holy shit," but basically saying, [03:00] "This is an incredible result." So a lot of people wouldn't have [03:04] expected this amount of progress in this amount of time, so this is truly [03:07] shocking. So you're now speaking as a person who is [03:12] dealing with the feelings you've felt over the last couple of days, having found this out. [03:16] So maybe also it'd be useful to go back to the, [03:20] the feeling you were having when you first saw [03:24] the report that, in this case, OpenAI dropped. [03:27] And the two of you happened to be on camera loosely on the same day, [03:33] maybe a few drinks in because it was a conference or a get-together. [03:37] Maybe, Ankur, can you speak to what you were feeling? [03:40] - It was an interesting night when the results dropped. Now, it was not entirely [03:44] unexpected because for weeks there'd been rumors about the two biggest [03:48] problems in my areas that were in that collection of [03:52] results, so the unique games conjecture and RL equals L. [03:56] There were even rumors about matrix multiplication, so really my first reaction [04:00] was, "Wow, it's all true. Thank goodness it's [04:03] finally out there because now people can actually go through it and process [04:08] what's happened and figure out what to do next." [04:10] Now, as to your earlier question, you know, what are we feeling? Well, for me, [04:14] it changes on a daily basis, but I think the biggest, most [04:18] overarching feeling is really feeling like you're getting pulled in two very [04:22] different directions. My work has always been about asking the right questions, [04:28] and to that extent, these tools are amazing for me because, you know, [04:32] they increase my ability to learn about other fields. It's like [04:36] having a world expert who's there to teach you about things in [04:40] your own language. Your ability to make connections is just magnified. [04:44] But, you know, one of the parts of my job that I really love the most is the mentorship [04:48] aspect. Not just training grad students and postdocs and seeing them do [04:52] great things, but having them come with me on these journeys. So the [04:55] part that pulls me in the other direction is the recognition that, [04:59] you know, the playbook and the blueprint for success that we had for so long is [05:05] now thrown out the window. [05:07] So it becomes a lot harder to advise students and postdocs [05:11] about, you know, what are things they should be working on that are gonna make an [05:14] impact, are gonna make a name for themselves. So I feel very excited, [05:18] but maybe more intellectually lonely than I felt before, [05:23] in part because, you know, a lot of times great results come [05:26] from asking very humble questions, at least for me. [05:31] And now there's so much anxiety and dread and [05:35] feeling that, wow, you would have to do something really momentous to [05:39] stand out in the sea of AI results, that it feels like [05:43] we've lost some of that patience in terms of planting the seeds from which [05:47] the great oak trees grow. [05:49] - Terence Tao, Fields Medal winner, wrote, "I am [05:53] horrified by the many person-years of ongoing patient and [05:56] deliberately slow research efforts, particularly by graduate [06:00] students and postdocs, toward many motivating problems in mathematics [06:04] being casually disrupted or destroyed by such a [06:07] release." Much as one cannot unhear a movie spoiler or a crossword clue, [06:14] one cannot explore a problem as profitably and richly once one is [06:17] aware of an existing solution, and I mourn the path [06:21] not taken and the opportunities lost in the frantic race to develop the [06:25] technology. So I mean, if we're being realistic, it's more [06:30] dramatic for a PhD student and a postdoc than, [06:36] than maybe the way you've articulated. I mean, you're saying it [06:40] quite calmly, but if I was a PhD student right now, I'd be [06:45] in a state of, like, shock. [06:50] Like, my whole life-- I was going through some of this. I know a lot of my friends are programmers. [06:55] They're going through this state of shock in last December when basically [06:59] most of the programming world switched to AI writing most of the [07:02] programs. So I would approximate it to a feeling of kind of a [07:06] loss of meaning, kind of terror, like dark stuff. Um, [07:10] we'll keep exploring that. Uh, Steve, what do you think? What were you-- what are you feeling? [07:15] - I would say most of my feelings are pretty hopeful. [07:19] Um, I think I'm an engineer. I like solving, you know, problems that [07:23] make technology more realistic for humans, artificial hearts and airplanes. And [07:29] our ability to solve real applied problems is going to get much, [07:33] much better. Our scope and reach is gonna increase, and so for that, I'm extremely [07:37] hopeful. But there is a lot of uncertainty, and I think, you know, some [07:42] of this fear of meaning, of losing control, of, [07:47] you know, just being kind of cut out of the process that we all [07:51] love being a part of it is a little scary. And [07:56] I don't think any of us would be surprised at either of the [08:00] kind of two paths that might take place over the next five or 10 years, where [08:04] we really are just dwarfed by you know, [08:09] other intelligence, or it kind of levels off, and this [08:12] becomes an incredible tool that we use for our [08:16] curiosity, for our human problem-solving. [08:19] And that uncertainty is massive. Those are two very different worlds. And so [08:23] uncertainty is difficult for humans. I think it's also [08:27] difficult for those of us who have chosen math. We chose math, you know, [08:30] probably because we liked rules and formalism and, [08:35] you know, uncertainty is uncomfortable. [08:37] I think the humility aspect is really important, though. This is a really good [08:41] opportunity for us to remember why we're doing this. Like, what is human purpose? [08:45] It's not to be famous. It's not to have a Fields Medal. You know, [08:49] solving interesting problems, helping other people, [08:53] being curious, just wanting to understand the world those [08:57] are really good things to kind of get back. Like, what do we [09:00] need to be satisfied humans? And I think that's a really interesting [09:04] moment for us to think about. Like, I've been thinking about it a lot lately. Um, [09:09] it is more difficult for young people, right? Like figuring out how to, [09:14] to find their way. But I actually have more faith in them than, than us in a sense. [09:19] - Do you think there's something to what Terry says about [09:23] it's harder to be motivated when you know a problem has been solved, [09:28] to explore that space when a problem's been solved like this by an AI system? [09:36] - Yes and no, in the sense that [09:40] I do very much agree that the path is as important as the [09:43] destination. Like, if you go on a road trip, the road trip is, you know, is [09:47] usually just as fun as getting there. [09:50] At the same time, if we found out that another country had [09:54] invented teleportation or time travel, [09:57] I guarantee we would reinvent it in the next five [10:01] years. Like, if you knew it was possible and you could reverse engineer a little bit, [10:05] we would be curious, we would dive into it, and we would figure it out. So knowing there's a [10:10] solution also is an incredible motivation to get there much faster. [10:14] - Let me ask the question this way. If this moment [10:19] marks the moment of the trajectory where AI basically, quote-unquote, "solves math" [10:25] or kills math, [10:27] can you make the case that that is in fact a trajectory, and then can you make the case that [10:31] that that is not a trajectory? Can you steelman both paths? [10:36] That this is a path towards which, like, mathematics [10:40] as we know it is done completely, [10:43] so that's gonna be a thing done by machines, and then what humans [10:47] do is something fundamentally different, or there still will be [10:54] something that looks similar to our current mathematics. [10:58] Can you make a, steelman both cases? Alex. [11:02] - Yeah. So, you know, you said two things, and I'd like to separate them. One is solve math, and the other [11:06] is destroy math. So, uh- [11:08] ... let's make that precise. What does solve math mean? If you think [11:12] back to when chess engines first came out and people were like, "Oh, this is gonna be [11:16] amazing. We'll play centaur games." So this is a human being plus [11:20] a computer system against another human being and a computer system, [11:24] and that lasted like 10 seconds- ... and then the computers got so [11:28] good that if you did anything but choose the top move that the computer [11:32] recommended, you would lose. [11:33] And so the human being's role in that game became, "Yes, [11:37] accept the first move. Yes, accept the first move." That's no longer fun, [11:41] and we don't see many games of centaur chess anymore, [11:45] right? And so in fact, I just had a conversation [11:49] with a colleague named Thomas Koberda where we axiomatized [11:53] our argument for what the future might look like, and one of the axioms [11:57] that I'm willing to even accept for the sake of argument is what I call the [12:01] solved axiom, and the solved axiom says, let's assume at [12:05] some future point, we're nowhere near there now, and I don't expect us to be [12:09] there necessarily. We'll see what you know, where the capabilities taper [12:13] off if, if they do. But let's assume the solved axiom which [12:17] says any mathematical problem that can be solved with human [12:21] interacting with AI could also have been solved by AI alone. [12:25] That's the solved axiom. So that, that would mean that really [12:28] there's nothing new for a human being to cont-- you know, you can say to an [12:32] AI system, "Okay, I have problem X, I wanna think about problem X," [12:36] but the only way that I will make progress, any progress that it'll make on problem [12:40] X, will just be press a button. The AI thinks about it. It doesn't necessarily solve [12:44] it, doesn't mean that it'll solve every problem. The Riemann hypothesis is not solved, [12:49] okay? That's what I really want the solution to. I just didn't think in my lifetime I would [12:53] see any kind of progress- ... towards it that we saw a few days ago. So [12:59] it's not that AI systems are solving every problem. They're solving every problem that [13:03] could be solved by humans interacting with AI. That's what that [13:06] axiom would be. So I'm saying, if we are ever in that world, [13:11] does that necessarily destroy mathematics? And my strong [13:15] argument is no, I don't think it destroys mathematics. [13:18] - If that axiom is true, what's the role of humans? [13:21] - So I don't think that mathematics, pure [13:25] mathematics, not engineering mathematics, not things that will, [13:29] cure cancer and, and solve, you know, jet-- everybody has a jet [13:33] pack or whatever other technological advancement you might imagine. Pure mathematics, you [13:37] know, we know that we're proving theorems just because we're curious about them. [13:41] I don't think the solution to theorems living in ones and zeros in data [13:45] centers is valuable to human beings. I think the value of pure [13:49] mathematics is that it lives in the heads of human beings. [13:52] - So just to, you know, add to this, I mean, I'm glad you came right out with the question [13:56] that's on everyone's minds. [13:57] So thank you for that. Um, I would caution against overfitting [14:01] to one objective function for this, right? [14:04] Because, you know, we used to think of ourselves as just the smartest things around. [14:08] You know, human mathematicians can solve, you know, reach the highest [14:11] treetops. But now we're faced with the fact that many of the problems that we [14:15] thought were, you know, quite a ways out have been solved by AI. [14:20] But maybe one of the consolations we can take is actually the fact that we're very [14:23] computationally limited compared to the way that AI is solving it. [14:28] You know, given the same finish line of proving the same theorem, [14:31] I can't really hold that many papers or ideas in my head at one time. [14:36] So in order for me to reach that same finish line, maybe I have to develop [14:40] more generalizable and abstractable ways of getting there [14:44] that then help you not just solve that one problem, but, you know, [14:48] explain connections or other directions you want to get [14:52] into, right? So it's always amazing when you look back on papers you've [14:56] written, and you can see these threads, invisible sometimes, that connect [14:59] different areas of work, because developing systems of ways of [15:03] understanding problems helps you for surrounding problems, not just to get [15:07] a Lean, formalized, you know, one theorem down. [15:11] - But the interesting thing here is those threads, what you're saying is sometimes [15:15] invisible, are quite visible to the AI systems now. It seems [15:19] like that's perhaps the actual very reason why they're [15:23] able to make so much progress, is they can extract the threads [15:27] and pull on them because they have the view of the so many papers. [15:31] - I think it depends on what you mean by threads. You know, for me, the reason [15:35] you prove results is so that you better understand the world around you. [15:39] Mathematics, to me, is a vehicle for abstraction and understanding things. [15:45] And you know, it's when you have a major result, what's [15:49] important is not just the truth of the statement, but what you've [15:53] learned en route to doing that. Usually, people's first question is, [15:56] "Okay, you proved something fantastic, but how did you do it?" And [16:00] then people, once they digest that new idea, that new definition, that new [16:04] way of looking at things, there are lots of other problems nearby that fall. [16:09] So somehow you're building structures that help you attack not just one problem, but [16:13] a whole sea of problems. [16:15] And, you know, it's not that we need fully formalized, you know, [16:19] results to do that. Sometimes we reason in very sloppy metaphors [16:23] that, you know, help us conceptualize the world around us. [16:27] - And how much of this is something Alex mentioned offline, attention is all you [16:31] need. How much of this is the human ability to shine a light on particular areas [16:37] that, quote-unquote, matter? [16:39] So the human ability to ask the right questions, the important questions. [16:42] - That's always been the case. So within this-- before AI, there's a-- [16:46] you can think of, like, here's the axioms, and just abstractly from the axioms, you can get to logical [16:50] deductions, right? There are theorems that we could prove from the axioms. And [16:54] there's a much, much, much larger space of theorems that we could prove than the [16:58] space that human beings ever actually care about. [17:01] And so, you know, so by, almost by definition, it's [17:05] like a tautology that the theorems that we'll get excited about are the theorems that [17:09] we care about as humans. [17:11] So maybe AI can convince us that there's some interesting structure. Maybe it can say, "Hey, [17:15] look over here. Let me give you this new definition or something." It hasn't done that yet. [17:19] - And sometimes the mechanism by which we select the things that we care about, [17:23] it could have an application. So Navier-Stokes is connected to fluids and all, [17:27] um natural systems. But it could also be a thing that some kinda mixture of beauty [17:36] and difficulty that just pulls at us. Maybe there's an intuition. [17:41] You know, the prime numbers are full of these really complicated problems, [17:46] That maybe don't have an obvious application in the [17:50] immediate sense, but it pulls at us somehow and maybe, [17:54] As a symptom of the pursuit of the hard thing, [17:59] Great applied mathematics and engineering and [18:03] solutions to real world problems comes out. [18:06] - So just, you know, as an example, right? A lot of times when I'm working on a problem, [18:10] I might have some, like, bigger goal that I want to accomplish, but usually what I [18:14] do is I try to find the simplest, most embarrassing open problem [18:18] that I still don't know how to solve. [18:20] You know, that aspect of like digesting it down to something, boy, how can we [18:24] not know this? It, you know, it feels like it's on the right track because [18:28] then when we can solve that example, maybe we're assembling the set of tools we [18:32] need to make attacks on other problems. So I do [18:36] think we're gonna have to become a lot more deliberate about, [18:39] you know, what do we care about? How do we use these tools from a [18:43] metacognition standpoint? You know, what exactly are we delegating to them [18:47] and what are we not delegating to them? And we're really at the beginning stages where [18:51] we're just trying to piece it together as we go along. [18:54] You know, on the one hand, looking for a proof is kind of like exploring this tree of [18:58] possibilities. You might have some intuitions based on some [19:02] other proofs you've seen about, you know, which directions to go in. Maybe [19:06] I should go down this branch, maybe I should go down this branch. But sometimes we [19:09] encounter, you know, points on the tree which have really high branching factor. [19:13] We just don't know which way to go down. We don't even have a guess about if we went [19:17] down them, what would we discover? [19:20] And now in this centaur world of math, we suddenly, you [19:24] know, have something else intelligent that we can interact with, right? And what is [19:28] the right way to use that tool to search for the beautiful theories, I think will be [19:33] challenging and evolving, but it's important to not lose sight of the fact that [19:37] we don't just care about theorems. [19:39] So I, you know, I'm a mathematician, I'm a theoretical computer scientist, but [19:43] I've also been reading a little bit more broadly, you know, about social [19:47] sciences just to sort of make my peace with all of the changes that have been happening. [19:51] One of the things that really stuck in my mind that I liked is there's something called the [19:55] doorman fallacy, which is very interesting. [19:57] So the idea is that you take a complex job [20:00] and you boil it down to its simplest, most visible task. So what does a doorman do? [20:08] I guess they hold the door for people as they're coming and going, right? And [20:12] when we lived in New York, we really enjoyed talking to our doorman. Uh, [20:16] you know, they were wonderful people, but, you know, if you digest it down to [20:20] just that simple job, you can think, "Oh I [20:24] can replace them just by a sliding door that detects [20:28] when people are going or coming," and you miss out on all the aspects of their [20:32] job that, you know, add a lot of value and are harder to articulate. [20:37] And I think that's a good way to think about mathematics in the age of AI too. What does [20:41] a mathematician do? [20:43] You know, the simplest and most visible part of the job is we prove theorems. [20:47] And, you know, theorems about conjectures that were open for hundreds of [20:51] years, like Fermat's Last Theorem, these are very visible things that [20:55] we've spent a lot of time popularizing. But I don't think that [20:59] that's all that a mathematician does. I like to think about it as a mathematician [21:03] is someone who thinks deeply, and a byproduct of thinking [21:06] deeply is, you know, proving old conjectures, but it's also [21:10] building new theories, building new ways of looking at old things, making [21:14] new conceptual connections. And these are parts that [21:18] you know, maybe AI will swallow up too, but we're not there yet at [21:22] this moment. So even though this is a momentous time, I don't think [21:26] it's the right time to hit the panic button. I think it's the right time to really [21:30] recenter in terms of what our values are and have these conversations as a field, [21:34] and remember that what we do really has a plurality of different aesthetics [21:38] and goals, and that when one thing becomes cheap, [21:42] like problem-solving for well-defined things, it makes the [21:46] other things that are orthogonal to it all that much more valuable. [21:49] - That's actually really beautiful. The doorman fallacy, going to Perplexity here, is the mistake [21:53] of reducing a person's job to its most obvious task and assuming that [21:57] automating that task replaces the value of the whole role. A [22:01] hotel decides, quote, "We're paying someone to open a door. An [22:04] automatic door can do that more cheaply." But [22:08] in fact, the doorman also welcomes guests and recognizes returning [22:12] customers, handles luggage and hails taxis, provides [22:16] security and discourages unwanted visitors, and [22:20] signals prestige and makes guests feel cared for. [22:23] - That was my favorite part. [22:25] I looked up what a doorman does, and boy, we have a lot of commonalities with [22:28] mathematicians. If I don't signal brand value, I don't know what I'm doing. [22:32] - Exactly. That's funny. But I think the task [22:36] for us humans in general, but also mathematicians, [22:41] is to come up with that list, like actually make more explicit exactly what it [22:45] is that humans do. And I haven't heard any of you [22:49] mention beauty. I apologize if I've mentioned beauty [22:52] excessively already. But there's something that mathematicians [22:56] are, from my outsider sort of fan perspective, are kinda [23:02] expositors, scholars of things beautiful, broadly defined. [23:08] - That's exactly right is until five seconds ago, there was a [23:12] near perfect correlation between the ability to solve [23:15] new deep theorems and what we actually want as human [23:19] mathematicians. And what we want is exactly what you said, [23:23] beauty and understanding and depth of thought, and [23:27] the fact that you can press a button and get what used to be our proxy, [23:32] without any effort, without any deep thought, and without any, [23:36] necessarily knowledge. Like we have people saying, you know, "Do [23:40] a breakthrough," and then putting whatever they find on Twitter and [23:46] and now, you know, what are mathematicians supposed to do with that? [23:48] - I would go one step behind answering questions and doing problem-solving is [23:53] asking the right questions. So like that is how I think you measure [23:57] good problem-solving is asking the right sub-questions, asking, you know, a [24:01] question that is interesting and can be solved and can be [24:05] interpreted. And so I feel like the value of asking good [24:08] questions has always been something we value in our [24:12] colleagues, and that's gonna become increasingly more valuable. [24:16] - I agree completely, and just to, you know, emphasize a point that was sort of [24:20] hidden in there [24:22] I think asking the right questions requires a lot of mathematical insight. [24:26] - Mm-hmm. Absolutely. [24:26] - So, you know, you'll see people make that comment all the time, you know, "What are we gonna value [24:30] going forward? It's asking the right question. It's giving great talks." [24:34] But, you know, I wouldn't minimize, not that you were, but just to be explicit, I [24:38] would not minimize the role in mathematical insight in both of those [24:42] tasks, because it's when we say that, you know, talks will become more important, [24:46] I'm not imagining that, you know, we're all gonna become Twitter [24:50] influencers- ... and TikTok whatevers. [24:53] - I hope not. [24:54] - I hope not. Because, you know, really what stands out [24:57] is having ideas that get adopted by your [25:01] community that end up becoming powerful over time. So, you [25:05] know, I wanna tie this back to something that Lex brought up earlier on, which is, you know, [25:09] he asked, am I being overly optimistic? Probably, right, compared to, [25:13] you know, how grad students view these things. But I think one of the things [25:17] that I'm most worried about in the near term is what's gonna happen to our systems and [25:21] institutions. [25:23] You know, it's not just hiring and tenure and all these things, but it's really [25:27] that when a grad student starts off their PhD, you know, they're [25:31] a new entrant to the field. They're trying to absorb what are the [25:34] techniques in their field, what are the important problems, and they're trying to [25:38] figure out how to make a name for themselves. [25:41] And what we used to do is we would give them a blueprint that was very simple to follow. [25:45] Take a problem people care about and improve it, and then [25:49] people will know who you are. You know, you'll get papers in top [25:52] venues. You know, doors will open for you. But now I think that [25:56] blueprint has changed in a lot of ways because there are so many [26:00] results that are happening right now that we're all just overloaded with it, [26:05] and I think it's okay to take time to rethink that blueprint. We still care about the [26:08] same things, like ideas that are gonna influence your community, [26:13] but they're just harder to operationalize in terms of some simple [26:16] rubric that we can give them. So I'm still optimistic. I just think it's gonna [26:21] take, you know, reworking a lot of our systems and institutions to do that. [26:25] - By the way, and so to, sorry, to go back to your point, I don't actually believe the solved [26:28] axiom. I don't think we're there. I can prove a theorem, but I don't [26:32] understand why the proof works. There's some magic that happens over [26:36] here, and until I generalize somehow the [26:39] phenomenon and understand it more deeply, then I'll see why the, the [26:43] proof works the way it does. Maybe that's one of the reasons why I don't actually believe the [26:47] solved axiom, because the things that we do are much more [26:50] exploratory and what constitutes a good question [26:54] beyond just, "Here's the Riemann hypothesis, make some progress on it." [26:57] - So I think you also brought up a really important point, this, like, [27:01] generality and generalization of our concepts. So related to beauty and mathematics, [27:06] I think one of the gold standards has always been how simple [27:10] can your result be and how much does it generalize. And, you [27:14] know, out of this infinite space of theorems that could be proven, [27:18] Finding some structure that is simple and [27:22] generalizes, like, that is, I think, still the gold standard. And unless I'm [27:26] mistaken, humans are still asking the questions. So, like, we, you know, prove [27:32] existence uniqueness of Navier-Stokes with AI, but [27:36] humans are guiding this. Humans are still asking the questions. And, like, I would love to see [27:40] the next breakthrough if you had a machine that can prove [27:44] any theorem you pose. I want it to start telling me what questions [27:48] I should be asking. [27:49] - Do you think it's possible that that machine just does a for loop- [27:54] ... and finds all the simple and generalizable, [27:58] basically all the E equals MC squared? [28:00] - That's happening today, actually. So, like, there are absolutely research programs that are [28:05] systematically, automatically discovering kind of minimal physical laws. [28:09] So this is less mathematical theorems, more physics governing laws and equations. [28:15] - So observing natural systems and trying to see in those natural systems- [28:19] ... if there's simple physical laws that model how they operate. [28:23] - Yeah, distilling it down to, like, the basics. So you can kind of [28:26] recapitulate known theories. If you observe balls dropping, you can, [28:30] you know, deduce gravity and F equals MA. If you observe fluids, you can deduce [28:34] Navier-Stokes. And then taking that to data, we [28:38] haven't written down simple equations: weather, [28:41] turbulence, solar flares, things like that. [28:45] And I'm assuming similar things are happening in math where, you know, there are [28:49] probably these, like, minimal pieces that can be deduced. [28:54] - Do you think there's a large space of simple and generalizable things in math [28:58] that could be discovered automatically? [29:01] - So my version of the solved axiom is that anything that I do [29:04] as a human mathematician, [29:07] I've tried on several occasions to sit down and say, "Okay, [29:10] you know, LLMs will never do this that I do." And I'm like, "Well, [29:14] why not? I can ask a question. You know, I can ask it a question." [29:18] I can ask it, "What questions should I ask?" [29:21] Right? There's sort of it-- Because it's such a generalized system, [29:25] uh, I'm not saying that it's good at all of those things now. But [29:29] my version of the solved axiom is anything that I can contribute as a [29:33] human being, it could also have contributed on its own. And so [29:37] what I really mean by that is in terms of value, what do I [29:41] bring to the community? It can't be I asked a great [29:45] question. It can't necessarily be in this world of the solved axiom, which [29:49] again, I don't believe. But even in that world, it should be something other than I [29:54] asked a great question, I proved a great theorem. And even in that world, [29:58] I think you need mathematicians, people with human mathematical expertise. [30:02] - Ankur, do you believe in the solved axiom? Where on the spectrum of, like, [30:07] how close we can get to it, [30:09] and in that spectrum, what's the role of the human, do you think? [30:13] - I guess I don't really believe in the solved axiom. I [30:17] think- You know, maybe it's half true in some ways, and figuring out [30:21] which half of it is true will be very interesting and challenging. So I think there'll [30:25] be major surprises and more to come in terms of what AI [30:29] can solve. But, you know, for me, when you're facing a new [30:33] technology that's really, really good, the danger is that [30:37] when you try to conceptualize why it's working, you do so by anthropomorphizing it. [30:43] So, you know, maybe one of the earlier lessons here is from deep learning. [30:47] So there are these amazing examples. You know, you can take neural nets, and you can [30:51] do human-level or better object recognition. [30:55] But in general, the way that it succeeds is less robust than how a human does, [31:01] Because there are all these examples like adversarial examples, [31:05] where you can add very slight patterns to natural images that can completely fool [31:09] these object detection systems, but not a human. [31:13] And, you know, I had a former colleague at MIT, Alex Madry, that had some really [31:17] nice work, which I really enjoyed. It was, um-- I think the title was something like, [31:21] uh, Adversarial Examples Are Features, Not Bugs. So they did the following [31:25] cool experiment. So you take a bunch of natural data, maybe pictures of cats and [31:31] dogs. You train a deep network to identify them very well. [31:35] You add some adversarial perturbations to them that, you know, [31:39] make to the AI system a cat look like a dog and vice [31:43] versa. So it's kind of think like a small little pattern that changes [31:47] how the object detector works. And then you create a totally fake data [31:51] set. You know, it's got pictures of cats with dog patterns on top, and you [31:54] label it as dog, and vice versa for cats. And amazingly, when you [31:58] train a deep network on this with all the wrong labels, it's gonna [32:02] learn something that does well on natural data. So somehow, when we [32:06] recognize what a cat and dog is, we do it, you know, in a way that we can [32:10] articulate in terms of you know, its legs and its tail and its shape in general. [32:17] But a lot of times when you know, deep networks or AI can do [32:21] things that they might be able to find trapdoors or backdoors to solving the problem [32:27] that use a lot of statistical relationships in the data that a human doesn't. [32:31] So to me, you know, it's important to not, you know, draw [32:35] this boundary, you know, very sharply between what you can do and what [32:38] AI can't do because you're gonna be wrong. [32:42] But that doesn't mean that I don't believe there will emerge some kind of [32:46] boundary, if only for the fact that I think we [32:49] over-anthropomorphize the way that they solve problems. And when I'm looking through [32:53] the OpenAI solutions, they just don't look like a human has solved [32:57] those problems. So to me, there still seems to be a very big qualitative difference, [33:01] and that's an opportunity really for collaboration and centaur [33:05] math more than it is, you know, [33:08] dread about the AI overlords. But, you know, who knows? Could be wrong next week. [33:13] - What's your intuition about how much of this is AI just [33:17] tying together a bunch of thousands of disparate papers [33:20] and pulling at the threads, sorry for bringing back the metaphor, but [33:24] the threads of the different papers and then just basically [33:28] articulating the proofs that were already there in the different papers and you're [33:32] making minor connections? Or is it actually developing [33:37] new mathematical ideas? Have you seen glimmers of [33:42] whatever that genius AlphaZero move in the game of Go? [33:47] - Yeah, move thirty-seven. [33:48] - Move thirty-seven. [33:49] - So I haven't read all seven hundred of their papers. The ones that I [33:53] have looked at, you know, some people speak about this dismissively like, [33:57] "Oh, it was in the convex hull." You do the moves that we know as, [34:01] as humans, and you just do them really well and do a really good tree search. [34:06] I wouldn't call it the convex hull. I think Tim Gowers had a really good way of [34:10] talking about it instead as the semigroup generated by all of these [34:14] things. Yes, it's the same notion of putting together known moves, but the space [34:20] of known moves, what it can accomplish is sort of much, [34:24] much bigger than this kind of derogatory [34:28] convex hull. So with-- in terms of the quasi-RH [34:32] the moves that it took are all things that were known to human beings [34:37] and there's a very long cascade of them that is kind of [34:41] shocking when you look at it and say, "Oh my God, that's so beautiful. How did you think to-- How did [34:45] you..." I'm anthropomorphizing it, of course. "How did you think to do this move after that move [34:49] and after that move?" And yeah, that does it. [34:53] - Could you articulate to the more general public the intuition, [34:57] the line between semigroup and convex hull, the derogatory convex hull, [35:01] and quasi-RH, of course, is the Quasi-Riemann Hypothesis, that one [35:05] of the big proofs. [35:06] - Yeah. Let me try to do it like this. You have-- think of like a balloon. [35:11] So the inside of the balloon is its convex hull. [35:15] And so we have like all of these results that-- all these ideas that mathematicians [35:18] have thought up, and then anything inside of that is some linear combination of it. [35:23] But it feels compact. It's like there's this kind of closed, [35:27] enclosed area, and within that area, like AI hasn't really made new [35:31] progress 'cause it hasn't moved the balloon out. It hasn't given us a new idea yet. [35:36] - And it's derogatory because the intuition there is we could have [35:39] done it, mathematician could have done it because it's already quite obvious. It's already [35:43] within the space of things that have already been thought about and developed. [35:48] - Yeah. So it has read every paper. It's exactly what Ankur was just saying. It has [35:52] read every paper. It has internalized all of [35:55] that. That's more than any human brain not even could hold, but has [35:59] time to read and learn and internalize. So in terms of [36:03] breadth, yes, it beats any human being. [36:06] And then in terms of depth, you know, has it really generated [36:11] a genuinely new idea? And you can argue that at least [36:14] in the case of quasi-RH, I would not say it's a genuinely new idea. It's a, [36:19] it's a beautiful combination of a lot of things that were very well known. There are a lot of people who [36:23] are kicking themselves that had some idea in that direction twenty years [36:27] ago, and two years ago, there was a big breakthrough on cubic theta [36:30] functions and cubic Gauss sums and, you know, that ended up being the path [36:34] that the AI took to the solution. But so to contrast that with a [36:41] semigroup, I think of a semigroup as like tiles. So you [36:45] know, you're just playing tiles on the board, and what you're allowed to do is whenever [36:49] you have a tile, you can put another tile to the left, to the right, to the front or to the [36:53] back. And then you can just start tiling things. And [36:57] that's like a semigroup action. And the question is, well, [37:01] how far can you reach with that? You can go all-- I don't know. You know? [37:06] You can go really, really far out with existing ideas might [37:10] prove a whole lot more than we as humans have been able to explore. [37:14] - So your intuition is more in the semigroup [37:17] direction, meaning like there is at least glimmers of expansion [37:25] versus linear combinations inside the convex hull. [37:28] - You know, how we choose to interpret what it's doing I think [37:32] is up to us. But throughout the history of mathematics, there have [37:36] been very seemingly difficult and deep problems [37:41] that then someone comes up with a three-page solution [37:45] that is very elementary and could've been explained to an undergrad and [37:49] could've been discovered twenty years before when the problem was first posed. [37:52] And had it been discovered twenty years before, it would be an [37:56] exercise in a textbook somewhere. And because it-- that solution wasn't [38:00] discovered, and really bright people worked on it for a long time and made incremental progress and made [38:04] the problem really famous, then by the time it was actually solved, it was a really big [38:07] deal. [38:09] That happens from time to time in mathematics. And what I said a year ago when I [38:12] wrote-- So there was this um, International Congress of Mathematicians this [38:16] summer, and I was writing my proceedings that they wanted ten months in advance. [38:20] So I said, you know, "What am I gonna write about math and AI now that will still be relevant, [38:25] Ten months from now?" And I think what's there is somewhat [38:29] evergreen. I stand by what I wrote. But one of the things that I wrote there was [38:32] exactly that we're going to discover that a whole lot more problems [38:36] that we thought were unreachable by current technology actually are reachable [38:40] by current technology if you can put it together in just the right way, and that's what [38:44] AI is amazing at. Again, someone could interpret that as, "Oh, so it's [38:48] inside, you know, these solutions were inside the convex hull." You wanna call it that, that's [38:52] fine. I think that's, you know-- It's not invalid. [38:55] - So I like all these metaphors, you know, also the balloon. So the important thing about the [38:59] balloon is the boundary between what's inside and outside, and what makes it [39:02] derogatory is the fact that, you know, you haven't seemingly changed [39:06] the boundary. But I-- you know, definitely these problems have [39:10] changed the boundary. [39:12] And, you know, maybe a more concrete thing is just that the types of things that people are [39:16] gonna work on after that drop of OpenAI problems is forever different. [39:23] You know, there were at least a couple of results there that are like, you know, [39:27] field-altering results that will change what types of problems you give to young [39:31] researchers, what areas you want to explore, because your conception of [39:35] the landscape is just that much different. [39:37] - So not just the-- because it expanded, but because the very [39:41] expansion changed the direction of the interesting questions. [39:45] - That's right. So in fact, you know, one of the problems which probably, you know, maybe it's the [39:49] easiest to explain what kind of effect it had on people is not actually from the [39:53] OpenAI drop. It was an Anthropic result that was written up by my colleague Virgie [40:00] Vassilevska Williams and then Josh Alman. So this was about the [40:04] 3SUM problem. And I'd say that there it has a major effect on [40:08] what's called fine-grained complexity theory because there was a big [40:12] surprise that, you know, a very basic problem that you give people in, like, [40:16] coding interviews can actually be solved asymptotically much [40:20] faster than the solution anyone knew, [40:22] and that was counter to what the community was betting. In fact, there was a [40:26] whole theory built up around the idea that if this problem really is that hard, [40:30] then all these other problems inherit its hardness. [40:34] So now, as a result of that problem being easier than we thought, you know, having a [40:38] bit better of an algorithm, you know, yes, that gives you a lot [40:42] new algorithms across the entire landscape. But it also [40:46] changes the character of the field because the goal there was to put all those problems [40:49] together and said that we should be studying them all together. [40:53] But now we're sort of back in the state of affairs where maybe the right way to study them is [40:57] separately after all. [41:00] So I think these are examples where, you know, irrespective of how [41:03] much, you know, deep new generalizable ideas we get out there, [41:08] even just knowing the, you know, truth of the theorem is gonna change [41:12] the way that people organize research in that area. And, you know, there's no [41:15] putting the genie back in the bottle on some of these problems. [41:18] - I mean, imagine one of the seven hundred was factoring in polynomial [41:22] time. That would absolutely change-- Everybody would know. [41:25] Like, you can't use your credit card on Amazon anymore. All of e-commerce is gone. [41:29] All of secure communications is gone in-- like that. [41:33] - Can you explain factoring in polynomial time? [41:35] - Yeah. So if you give me two very large numbers, a [41:39] million digits each, I can multiply them together and do [41:43] that in a split second. If you ask me to go backwards, [41:47] say you took two large prime numbers, you multiply them together, you give me this, the product, [41:51] and you ask me, "Figure out which two numbers..." You know, just factor. Don't we, don't [41:55] we ask our kids to learn how to factor in school? You know, it's a lie. [41:59] We do not know how to factor. And the fact that we don't [42:03] know how to factor is why we have the capacity, the ability to [42:07] communicate with one another in all kinds of secure ways. So all of [42:11] modern-- basically the modern world depends on the fact that we don't [42:15] have an algorithm for taking a large number and pulling its prime factors out of it. [42:21] - So a lot of e-commerce does because that's the basis for a lot of the schemes [42:25] in public key encryption. But, you know, famously, there are quantum [42:29] algorithms for factoring, so- [42:31] ... there's a lot of emphasis these days on, you know, finding post-quantum [42:36] you know, bases for cryptography too. [42:38] - Yeah, lattice-based methods and- [42:40] Yeah, we're already concerned about what will happen when we have quantum computers large enough to, but we still [42:44] have a little bit of time before quantum computers get large enough [42:48] to make this an actual problem. But there's no [42:52] theoretical reason. I mean, okay, factoring's probably not NP hard but [42:57] You know, I mean, let's say, let's say prove that P doesn't equal NP. Maybe that doesn't-- [43:01] That would be amazing, obviously, but that doesn't if it proves that P [43:05] equals NP, that would be completely [43:08] - That would be very interesting. [43:09] - That would be very interesting. [43:10] - I'll definitely ask about more broadly in the context of complexity [43:14] theory, but presumably OpenAI or any other company [43:18] would not drop that result carelessly. [43:22] - That's the kind of thing that should go through, yeah, the government and- [43:26] - Sometimes certain mathematical results quite obviously [43:30] lead to dramatic impact on the world. Other [43:34] results might take maybe months to digest, but that one would just, [43:40] yeah, would affect the economic systems of entire nations, so. [43:44] - How many mathematicians does the NSA employ? [43:48] - Thousands. [43:49] - Thousands, like something like this. [43:51] I've always done this thought experiment, you know. What, [43:54] what might they have that they just wouldn't tell us about? And this was ten years ago, you [43:58] know, that some [44:01] evidence showed that collaborative teams of mathematicians could solve [44:05] much, much harder problems than any one individual mathematician if they, like, [44:09] broke it down in the right way. So I don't know. Do you mind if I just revisit the- [44:15] - Yes [44:15] - ... the balloon for a second? 'Cause I actually like the semigroup [44:19] analogy better than the convex hull, but I don't [44:23] think it's as it should be interpreted as derogatory as [44:26] it is even if it was just that these machines are [44:30] interpolating human knowledge rather than extrapolating. [44:34] Because even if there is this convex hull of ideas and things that could [44:38] be pieced together from bits that humans have invented, [44:43] we're not doing that fast enough or, you know, there's a [44:47] lot of opaqueness, and it's-- this balloon is not see-through. [44:51] And so even if you can make this convex hull much more transparent, kind of [44:55] like lift the fog, give mathematicians radar so they can see what's, you know, [44:59] over the horizon, [45:01] that's pretty incredible, and I don't think we should be interpreting that as you [45:05] know, derogatory at all. I do think it is more, [45:08] more extrapolatory than that, though. [45:11] - Yeah, we should say probably the size of the convex hull. We don't know [45:16] actually how big it is. [45:18] - Well, that's part of the point. We don't know how big it is. We don't know what's next to each other. [45:21] Like, there's so much fog of what we know. Like, And that's exciting to lift that. [45:29] - But human beings do make new definitions, you know, bring new [45:32] concepts in, and that fundamentally feels [45:36] different from combining existing ideas and repackaging them. [45:40] When human beings do that, and they do that from time to time- [45:44] ... it really feels like, okay, there's a new boundary point, [45:48] and now what's the convex hull of that, of what you could prove? So [45:52] it's like you can take the balloon, and there are ways to pull on it and- [45:55] - And we know when it expands. Like somehow- [45:57] ... we have an intuition for when it's getting bigger. [45:59] Like, you can recognize that. [46:00] - In many ways, the archetype of what the researcher looks like is gonna [46:04] change because of this new technology too. [46:07] You know, I'm always amazed that when you look back historically, how much [46:11] very different fields of math really spawned from similar ideas. [46:17] And a lot of those connections drove the development in the beginning, but over time, [46:21] they become a bit more specialized. And I think certainly the way that we [46:25] train graduate students, you know, we'll [46:28] need to make sure that they're not just overspecialized in the same way. [46:32] You know, I don't know what the future of hiring will look like. I don't know what the future of journals will [46:36] look like. But I think people who think broadly about, [46:40] you know, a bigger swath of mathematics will be in the [46:44] best position to benefit from these technologies. [46:48] You know, that would be my guess. Can I ask a almost silly-sounding question? [46:52] But so a large part for many decades of research has been [47:00] the paper. If you were to guess in the years to come, how does that [47:05] change how a human contributes their idea, maybe [47:09] collaborate, how a team of humans contribute an idea into the public space? [47:14] - So, I mean, I don't know about in pure math, but in engineering and [47:18] some of the applied sciences, we already have a pretty [47:22] rapidly shifting view of what an academic contribution is. So [47:27] if you write code in part of a larger infrastructure that [47:31] is reusable, that, you know, supports larger projects, so think of [47:35] LIGO or some of the big, you know, billion-dollar [47:39] astronomy projects or, you know, cancer research projects that [47:42] require hundreds of people and huge resources already, [47:47] it's not necessarily one 15-page PDF. You [47:51] know, there's lots of different pieces, code, data, [47:55] reusable bits that can be fit into a larger framework. And [47:59] in my mind, that is kind of obviously even more [48:03] relevant with AI, is having, you know, a little workable [48:07] piece or widget or API or code [48:11] base or data set that can be leveraged by the community or [48:14] by AI is increasingly valuable, [48:17] even more than just, like, one result or one paper. And I don't know [48:21] how that looks in, in pure math. [48:23] - Can you comment on your view how everything's changing? Even just observing [48:28] the insane evolution of even arXiv, [48:33] which is similar in some ways to what's happening on GitHub, for example. A lot of people [48:37] in the open source movement are really struggling to know what to do. Some are predicting [48:41] the death of GitHub. [48:42] - Why? Just 'cause there's so much code being added to GitHub- [48:46] ... that it can't sustain the growth? [48:47] - So one of the things that makes the open source movement successful is there's a kinda [48:52] one or a few folks that really manage each repository, the [48:56] human beings, and they are, um- [48:58] - psychologically and on many levels are becoming overwhelmed by the, all the pull [49:02] requests. So it becomes impossible to manage and they get [49:06] exhausted and there's this whole feeling. [49:09] The open source movement, maybe similar to the academic environment, [49:13] requires the humans to be motivated. And if you get worn out by the just similar [49:21] the ideas that Terry Tao mentioned, [49:24] that you just quit. And so that's a failure mode that you want to kind of avoid. [49:28] - So journals were already in math under very, very severe strain [49:32] from the number of things before AI, having nothing to do with AI, just [49:36] there was, you know, more people producing more knowledge, sending it to journals, and [49:40] journals had to referee it, and referees are exhausted and being asked to, [49:45] work really hard to understand these theorems and check them. [49:49] The only thing worse than that system, I think, is not having that system. [49:53] Because the way that, like, our mathematical epistemology, the way that we [49:57] arrive at truth, is deliberately slow and careful and precise. Go back [50:03] to Perelman's solution of the Poincaré conjecture. He puts these [50:06] papers on the archive, [50:09] and then people around the world are, you know, shocked by this, and so they invite him to come [50:13] give lectures. He gives he comes to Stony Brook, he comes to NYU, he comes to Princeton. [50:17] He's going all around the country and the world, and [50:21] he's giving three-hour long impromptu [50:24] explanations of, of, you know, maybe an hour is, is him explaining what he's doing, and [50:28] the next two hours are people saying, "On page two [50:32] hundred and forty-seven, Lemma four hundred and twelve, I don't understand how you [50:36] do this." And then he goes to the board and gives them a half-hour lecture on how he does that. [50:40] So the entire thing lives in his mind. [50:43] He's done the best he can to communicate it on paper, and then it's up to [50:47] us, the community, to go back through it very carefully, very [50:51] in a detailed and precise way and agree, yes, [50:55] this is a real proof of the Poincaré conjecture. And that took years to do [51:00] and a whole lot of human effort. So coming up with a [51:04] solution is only the first step in the process, and that process has to be deliberate, [51:08] and it has to be community-driven. And so that's what the journal [51:12] system is trying to accomplish. [51:13] - So you make a really interesting point and it kinda [51:17] highlights this traditional view of a pure mathematician as this, like, [51:20] lone genius who, [51:23] you know, solves this singular problem and then goes explaining it to the world. [51:29] In many other fields, we have moved past the [51:32] point where we can't individually advance, you know, [51:36] circuit design or aircraft design or protein folding or any, you know, [51:40] astronomy, and we have to work together in these billion-dollar collaborative [51:44] projects. And again, I'm an engineer, so I'm thinking leadership is obviously [51:50] the thing that a community needs when it's having this kind of crisis of [51:54] organization and individuality moving towards a more collective model. And that's [52:00] interesting because I know so many mathematicians, and, like, [52:05] by kind of organization, this is a very [52:08] independent group of people who don't like being told what to do, who don't [52:12] like being told what to study, who are very independent by [52:16] their very nature. You know, we like solitude and working [52:20] alone. That's why we, you know, spend so much time doing it. [52:23] - I think that's a really good point. I mean, it's interesting just to see the [52:26] differences in working styles, even between math and physics, you know, with things- [52:31] ... like the Large Hadron Collider. So I mean, let me take a contrarian [52:35] view for these things, which is you know, you should never let a [52:39] good crisis go to waste. I agree the journals were [52:43] overloaded, the conferences were overloaded. It always strikes me that we're [52:47] doing so much work, you know, to review these things, to handle [52:51] all these things, and now we're faced with the task that [52:53] someone's gotta review all these AI-generated papers. [52:57] And some of them are important, but, you know, it's really hard to get your community on [53:01] board with doing that. You know, I don't know what's gonna happen, but I would really [53:05] be in favor for finding new models, not to replace the [53:09] old ones, but to relieve some of the pressures that have come about. [53:13] So, you know, I'd love to see models where, you know, maybe we take a more [53:17] collaborative effort to verifying some of these AI [53:21] proofs, right? So some of the OpenAI results that have [53:24] dropped, you know, improvements on Fast Fourier Transform. [53:29] You know, there's been tweet streams with people improving the constant. [53:33] I don't really know what to make of that. You know, I'm not even on Twitter. [53:37] You know, and it seems to me that we need to figure out other modes of [53:41] working, you know, for organizing what are the state of all the [53:45] improvements, but also how do we organize and tame these ideas? [53:49] So it's kind of striking that, you know, the way that we think about publications is you [53:53] work hard on your draft, you post it on arXiv, and that's your draft. [53:58] You know, you might update and create a new version at some point or a third [54:02] version, but the draft belongs to you. [54:06] It's not really up to the community to add notes. "Hey, I think you can do this argument [54:10] differently. Hey, I think this is simplified this way." But maybe there are some [54:13] kinds of things that we need other models for, you know, [54:17] allowing crowdsourcing for our understanding to evolve. [54:21] I'd be all for that. You know, the other thing is I love giving [54:25] research talks. You know, one of my favorite parts about giving research talks [54:29] is you can take all of the messy complexities of the paper, the things [54:33] where you just knew you had to dot the I's and cross the T's, but [54:37] that wasn't really what excited you. There was some key idea, and there [54:41] was some special case that really illustrated why that idea was powerful. [54:45] And that's the part you really want to explain to people, because when asked, someone asks [54:49] me about Lemma 57, I don't even remember it, right? And [54:53] maybe, you know, this idea that uh, a paper [54:56] is just this monolithic thing designed to be [55:00] you know, written down and completely formalized. Well, now we can [55:04] trust Lean formalizations to some extent for some of the [55:08] technical building blocks, and that can allow us to rework [55:12] the way that we do the writing in the first place, so we can put more of the emphasis on the work [55:16] on, you know, the promulgation of those ideas. [55:20] - The Mathlib model is exactly what you were describing, where instead of a paper, you have a [55:23] PR, you have a pull request. The maintainers and the reviewers will [55:27] go through the pull request. They'll make suggestions here and there. I mean, the [55:31] referees make suggestions, you can improve this and that, but it's not a public process [55:35] in a journal. In a GitHub open [55:39] workplace, you can see exactly who contributed what idea [55:43] that simplified an argument or you're, you know, you're reusing this... You're reinventing [55:47] this idea that already exists in this part of the library. You should just quote that and [55:50] refactor this part, whatever. So and [55:55] Mathlib working formally allows one for the first time. Like, one of the [55:58] problems with mathematics is that we don't [56:02] have a perfectly well-written specification [56:06] to our theorems, our technical lemmas. There's-- If you open up a random [56:10] page in a random math book and read a lemma, there's so much context- [56:15] ... that the author-- 'cause it's so annoying to write something that [56:18] fills in every bit of context. So we just assume you've read the previous [56:22] sixty pages. Now I can just tell you what the next step is in that lemma. But then when you try [56:26] to-- That means you can't just take it off the shelf the way you can a part, [56:30] a physical part when you're building a rocket. [56:33] And so if all of mathematics is done in the future [56:37] formally, then the specifications are built into the [56:40] object. You know you're not using it in the wrong way, 'cause you can't use it [56:44] in the wrong way 'cause Lean won't let you, [56:47] and that is the sort of crowdsourced version of mathematics. And that was the dream [56:50] of Mathlib- ... when it was started ten years ago, [56:55] was to run mathematics more like a science. [56:57] - As a quick aside, can you, [56:59] mention what Mathlib is and Lean, just 'cause we mentioned it a few times for folks? [57:04] - Sure. So mathematics is, in the ideal, a formal [57:08] process starting with an axiomatized system and then logical [57:12] rules of inference from there. So if people think of -- I don't know, do people [57:16] still learn Euclid's geometry in high school or something? That's supposed to be [57:20] the kind of model for kids to start seeing [57:23] proofs for the first time and seeing an axiomatic structure. [57:27] And you can take that idea and write it in software [57:31] and say, "Here are the actual axioms of mathematics. Here are the rules of logical [57:35] inference." And Lean, there's a long history of this kind of software. It's called [57:39] interactive theorem provers, where you're writing line by line in a proof, and the [57:42] computer says, "Yes, this line does follow by logical inference from the [57:46] goal state on the previous line." So Lean is software that allows you [57:50] to check all kinds of things. It can check the specifications of your code [57:55] align with what you, [57:57] what you are trying to define. That's what it was -- what it was invented for was bug-free [58:01] code. And hopefully, in the future, we will have bug-free code because AI [58:05] systems are really good at finding bugs in code written by humans, so [58:10] that's-- it's becoming ever more important for that application. But [58:13] another application is to check mathematical results- [58:18] ... and allow you to write a mathematical result in a very formal way. And so [58:24] now Lean itself is just the hardware that checks the logical steps. You do not want [58:27] to start every math project by defining what the natural numbers are, [58:31] using zero and the successor, you know, the Peano axioms of the natural [58:35] numbers. So it's nice to have a large general [58:39] library that's extremely well maintained, extremely well [58:42] organized, and so on. And that's what Mathlib is. Mathlib [58:46] is the Lean library for formalizing [58:50] mathematics in the Lean ecosystem. And again, Lean is just one [58:54] interactive theorem prover. There are lots of others. It's the one that so far has been [58:58] winning the math wars, if there are any math wars. It's a competition of [59:02] two people or something. [59:03] - And it's the technology used in part to show that [59:08] the OpenAI release of all the different proofs, that the proofs are correct. [59:13] - That's right. It has become now standard to, [59:17] because autoformalization has gotten to the point where AI systems can not [59:21] only prove a theorem in natural language. You can't trust their natural language proofs. [59:24] These are stochastic processes. [59:27] And so the fact that they're saying things doesn't mean that they necessarily have [59:31] a model of mathematics in their heads, whatever that means, the same way [59:35] that we do. But that's okay 'cause they'll write a Lean -- Lean code [59:39] that builds on top of Mathlib, builds on top of the Prime Number Theorem Pro-- Plus [59:43] project or whatever, and checks that it has [59:47] indeed solved this problem down to the axioms. [59:50] - So this is where I'll just take the tangent, and the thought experiment may be a bit [59:53] philosophical at this moment, but what if you have that black box [59:57] that gives you proofs that you can't understand? Let's say the more interesting case [60:01] where, not because it's so big, but because the ideas just [60:06] don't make sense to you, to any of the world expert mathematicians in that field. [60:11] What do you do with that? [60:13] - So the only way, sort of almost by definition, because the [60:17] logical rules of inference are written down, and there, there's [60:21] only a few of them. There's a handful of them. The axioms, there's a handful of them. [60:25] So the only way that you have mathematics that's not understandable is scale. [60:30] - Let me add a contrarian take. [60:31] - Yeah, please. [60:31] - So one of the things I also wonder is what would unformalizing math look like? [60:37] You know certainly formalization helps tremendously that we [60:41] have confidence that the results are correct. But much of the way that we [60:45] understand the results ourselves is really at some unformalized [60:49] level. You know, what is an interesting special case? What, you [60:53] know, where does this come from, you know, this technique you're using, right? [60:57] And, you know, maybe that's what that world would look like. If we have these black boxes, we [61:01] would try to tease apart and figure out how to ask them easier [61:05] questions to develop some kind of system of understanding of [61:09] how they actually got to the answer. [61:11] - Yeah, so there are really two different things that we conflate when we write a [61:14] paper. One is the engineering of the proof, and the other is the [61:18] communication of that to someone else, and they're almost at odds with one another. [61:22] I had this experience-- So I taught real analysis last fall, and you open up [61:26] any real analysis textbook, and this was my experience when I took real analysis. [61:28] The first couple of lectures are what are the real numbers? You know, we have to [61:32] define them or construct them as equivalence classes of Cauchy [61:36] sequences of rationals, whatever the hell all that means. It takes, like, two weeks for a first, [61:40] semester course in real analysis to get there, and then we have the real number line, [61:45] and now we can start doing calculus, you know, with formal epsilon and delta [61:48] definitions, yada yada. And it always struck me as bizarre that [61:52] we do this because before the class, I knew what the real numbers were. It's this number [61:56] line, and then we spend these two weeks talking about [62:00] Cauchy sequences of rationals, and then on week three, they're, [62:04] they're a number line again. What the hell was the purpose of that? [62:07] And so it's because the way things are written or had been [62:11] written were very engineering. You need to construct the real numbers before you go start using [62:15] them. But the [62:16] real numbers, human beings used the real numbers long before we understood [62:20] that they needed to be constructed formally. So what I did in my class is I jumped [62:24] straight into epsilon delta calculus. We did, you know, six weeks of [62:28] calculations until we got to a point where [62:32] wait a second, you know, if here's a theorem, it's true in the real numbers. There's some [62:36] convergence, something called Bolzano-Weierstrass theorem. You have a sequence of real [62:39] numbers that are jumping around, but in a fixed range. There has to be some subsequence that [62:43] converges to a real number. [62:45] Well, that's not true in the rationals. Why is it true in the reals? What [62:49] are the reals? So you don't start asking that question until you hit a [62:53] problem that needs to be solved. So the way I taught the course [62:57] was sort of backwards from how it needs to be taught if it's an engineering project, [63:01] 'cause you can't just start talking about the real numbers until you've constructed them, until you know what they are. [63:05] Well, that's correct if what you're doing is building up the theory. But if [63:09] you're telling a story, then you absolutely can just start with the real numbers. [63:12] You have an intuition for what that is, and until your intuition breaks, you don't need [63:17] the resolution of how you actually construct them. So I think the way that we write [63:21] papers in the future is gonna be very different, where we spend a lot more time just [63:24] talking about the story of what have I learned in this process, [63:28] what are the interesting-- Uh, here, here's, you do the standard thing here, here, and here, [63:32] but then here, you're sort of stuck, and where do you go? And here's an idea for [63:36] why you go left and not right and not up and down or whatever. And there's a Lean [63:41] formalization that accompanies this story that lets you know that the engineering actually [63:44] works out. 'Cause you can't just tell stories. If you just tell stories, you know, math [63:48] is not music where you can tell [63:52] a beautiful story, and it's okay that you have parallel fifths moving somewhere, and, you know, you've broken [63:56] some rule, but Bach does it all the time, so. [63:58] - So I think that's a very important point, just to sort of emphasize the point [64:02] that Alex is making. You know, there's this view of mathematics which I don't believe [64:06] in, where it's just this rising tide of theorems. You know, how can [64:10] AI be bad? It's contributing to this rising tide. [64:14] But, you know, really, mathematics is not just about the number of theorems we [64:18] have, far from it. It's also about that story of fighting [64:22] with problems, because when you fight with a good problem, it fights back at [64:26] you, and you learn something about it. It's really about the system of [64:30] beliefs and abstractions you come up with en route to, you [64:34] know, when you're fighting this problem, you discover some more general meaning that applies [64:38] more broadly. [64:39] And I think it's a great opportunity to sort of disabuse ourselves of this idea that [64:44] we're just you know, axiomatizing things until we get to the next theorem. You know, [64:50] telling that story of how we actually got to these problems, what the [64:54] parallels are to other problems, is now, at this pivotal moment, more [64:57] important than it ever has been. [64:59] - Yeah, so to get all the way back to your question about, you know, is it a matter of scale, well, [65:04] Yes, it's just a matter of scale, but exactly to Ankur's [65:08] point, the way that we think about mathematics is about stories, and it might be that [65:12] we don't understand the story of this. You can follow a proof line by line and not [65:16] understand what's really going on. [65:20] You might say, and you just ask some other LLM to explain to you the story if it [65:24] can come up with such an explanation. But that's what we [65:28] want. We want stories. We want understanding. [65:30] - When you fight with a problem, it fights back. [65:37] I think there's a deep truth to that about all the great pursuits [65:41] that humanity has taken on. And on that topic, [65:45] I would like to talk to you about some of the greatest fights of all [65:49] time with some of the hardest problems in mathematics that have fallen. Um, [65:53] but first, we'll have to break if it's okay. So Alex, you're a number theorist. Um, [66:01] maybe it's fair to say the central problem in your field is the Riemann hypothesis, [66:05] and you showed a lot of excitement on the aforementioned Twitter/X [66:11] about the quasi-RH, the quasi-Riemann hypothesis proof. Can you explain that? [66:18] Can you explain what the Riemann hypothesis is, and can you explain [66:22] the nature of your excitement about it? [66:25] - Sure. There's sort of a one-minute version of the Riemann hypothesis- [66:28] ... or a five-minute version or a thirty-hour version. [66:32] I don't know how far you wanna, you wanna go. [66:34] - Let's start pulling at that string. [66:36] - So we know what the prime numbers are. People are interested in the distribution of the prime numbers, [66:40] and what Riemann discovered is that there's a function, this is totally [66:44] crazy, that there's a function that takes complex numbers as inputs and [66:48] has complex numbers as outputs, and that function [66:52] governs the behavior of the prime numbers. It's called the Riemann zeta function, [66:58] and by the zeros of the zeta function, we mean complex inputs [67:02] where the output is the number zero in the complex plane, so the, [67:06] the origin in the complex plane. [67:08] And the question is, where do these, where are these complex zeros? What inputs, what [67:12] complex number inputs give you a complex number output of zero when [67:16] you apply this zeta function? [67:19] Why in the world should that-- What, complex numbers? Are you-- What are you talking about? [67:23] We have these discrete numbers that aren't divisible by anything else [67:27] sitting in the natural numbers on the number line, and now you wanna talk about [67:31] functions from a complex space to a, to a complex space. Um, [67:35] that was the great breakthrough. The great breakthrough was to figure out how to do [67:39] signal processing on the prime numbers. [67:42] If you think about, you know, the reason people can hear [67:46] you, your podcast is that my voice box is producing [67:50] pressure waves of sound. That's the natural space. [67:55] Then that goes into the microphone. The microphone analyzes it, does the Fourier analysis [67:59] on it, converts it into amplitudes and frequencies, [68:03] and it's just a bunch of numbers, stores those numbers or sends those numbers, whatever. [68:06] The speaker then takes those numbers and reconstructs back out what [68:10] sounds to people like my voice. [68:13] The same thing is going on-- That's what the zeta function does for the prime numbers. [68:17] It takes the prime numbers, you can think of the prime numbers as particles of air molecules [68:21] wiggling around, and it does a signal processing analysis on it [68:28] that can recover the same prime numbers back. [68:31] And the frequencies and the amplitudes correspond to the [68:35] zeros of the zeta function. The zeta function has that information in it. [68:39] And the Riemann hypothesis, so the amplitudes [68:43] of all of these, you can call them Riemann waves, [68:48] and the superposition of those waves recovers the distribution of the primes. [68:52] And all of those amplitudes are supposed to be exactly [68:56] the same. Like, the orchestra that plays the, [69:00] the music of the prime numbers has every single instrument sounding at [69:04] exactly the same amplitude. That's what the Riemann hypothesis predicts. [69:09] - So can you-- Actually, we know what the prime numbers are, but [69:13] maybe going into the space of philosophy and the romantic [69:17] notions of aesthetics why are they so interesting [69:21] and important and central and beautiful at the core of mathematics? And, [69:28] connected to why the Riemann hypothesis is exciting is why is it so [69:32] difficult to figure out if and what the pattern is in the prime numbers? [69:37] - Let me start with your second question first. Why something is difficult [69:41] I only know how to answer once I know what the answer is. [69:45] - Once it's made simple. [69:45] - Right? Maybe it's not difficult. [69:46] - Okay. [69:47] - Maybe it's not as difficult as we thought it was. I did not think I would see a proof of the [69:50] quasi-Riemann hypothesis in my lifetime. So the fact that I saw it on Tuesday, [69:56] was extraordinarily shocking. [69:59] - What did you do the moment you saw it, like, in 30 [70:03] seconds after? Like, what was your-- Do you remember, like, what- [70:07] - Yeah, I remember where I was- ... the second I saw the drop. So we [70:11] were at this National Museum of Mathematics Gala- [70:14] ... in New York City. And so I had, you know, a whiskey in one hand, and, [70:18] I kept refreshing. I was talking to people, socializing, but I also kept [70:22] refreshing 'cause I knew it was dropping around six o'clock Eastern Time. [70:27] And then the drop hit and, in fact, I was talking to [70:31] there was a Quanta reporter there and we were just chatting, "Oh, yeah, they're gonna [70:35] drop these, you know, 300, 400," we didn't know how many, 700 it [70:39] turns out problems. And I was like, "Oh, wow, that's really cool. It [70:42] dropped." So I'm looking at my phone, and, like, the second thing I see is [70:46] quasi-RH. And I'm like, "What do you mean, [70:50] quasi-RH? You don't mean, like... For what?" And then I look, and it's, [70:55] it's the zeta function, and it's all Dirichlet functions, so they've solved no zero, zeros. [70:59] And I'm like, that can't, that can't be what's in this drop. That just-- I, [71:08] I could not compute- ... in that moment what I was seeing. [71:11] - That speed of progress in that little time, yeah. [71:14] - It was so far outside my model of what was possible. [71:20] And this is-- It's been a month of things happening way outside [71:24] of my model of what was possible. The Fermat's Last Theorem formalization, [71:29] I was sure we were three years away, at least, from being able to do [71:32] something like that. [71:33] - So, maybe you can elaborate on that. It was in Lean, it was formalized [71:37] as... How big was the Fermat's Last Theorem in Lean? [71:41] - Like, 13 million lines. [71:42] - 13 million lines. Okay. [71:43] - So I just-- Two days before, I gave a colloquium in the Princeton Math Department where [71:47] I explained in painstaking detail what Lean is, what Mathlib is, but also [71:51] that AI systems today, you know, there's this Mathlib library [71:55] where everything is kind of perfectly done, exactly the right levels of generality, enough-- [71:59] There's no-- There's not supposed to be any code that's reusing [72:04] code that's reproving theorems somewhere else. Everything is the most general, [72:08] the most abstract, and what AI systems are capable of doing is building, like, one [72:12] or two layers on top of that. But they're, what they're doing is jerry-rigging. They have [72:16] not been trained yet, I'm not saying they can't do it, but [72:20] they're not good at it right now, at what I call canonization, which is [72:23] taking kind of sloppy code, refactoring it, figuring out exactly where in [72:27] the library it belongs, what's the canonical version of this statement, of this [72:31] lemma, and then everything, every proof should just be a combination [72:35] of you apply this lemma from over here, you apply that lemma from over there, put all these [72:39] things together, there's your theorem. And then it's really clean, it's really beautiful, very elegant. [72:43] That's what Mathlib is going for. And what I said in my colloquium is, where we [72:49] are today is that you can get entire textbooks formalized as long as they're in the Mathlib [72:53] halo. [72:54] As long as they're, like, one, maybe two steps removed from things that have already been [72:58] done in Mathlib. But when you start, and this had been my practice, my experience had been, [73:02] when you start asking it to do things that are, you know, three or four steps removed from, [73:07] the library, the way that it came up with a definition of [73:11] this object and the way that it came up with some other definition of the same object, 'cause it was doing it [73:15] for two different purposes, then they clash. You can't stitch these things back [73:18] together. Like, you really need the library, Mathlib, to grow- [73:22] ... in order to get to a place where you can [73:26] do research manuscripts, you know that are, that are really [73:30] using, synthesizing tons of different areas of mathematics. [73:33] - Which is why you would expect Fermat's last theorem to take-- because it's so far [73:37] outside of the halo of Mathlib, it would take quite a bit of time. [73:40] - Exactly right. So Kevin Buzzard had a million dollar [73:44] grant to spend the next five years, or I th- maybe he's already a year or two into that [73:48] grant, to reduce the formalization of Fermat's last theorem [73:52] to things that were known in the 1980s. [73:54] So he wanted to do that last step, the, those seven years of Wiles' work, [73:58] and Wiles Taylor, to finally finish you know, Fermat's [74:02] last theorem, but built on top of things that we knew would be extraordinarily [74:06] difficult. He wasn't gonna do Langlands' Tunnell, he wasn't gonna do Mazur. There were, there were all kinds of things [74:10] that were expected to be even much more difficult than what these five years would [74:14] take. And everything I said in that colloquium was wrong. [74:19] And I think my colloquium was on a Wednesday. On Friday, there was a drop of thirteen million lines [74:23] of code from the axioms, Fermat's last theorem is proved. [74:27] The statement of Fermat's Last Theorem is completely elementary. There's no semantic alignment to [74:31] check. You just glance at it. Yes, that's the statement. It's-- [74:35] Lean says it's sorry-free. [74:36] - Can you explain how it was done? [74:38] - How it was done. So it was Anthropic, and how it was done was AI [74:42] systems just got really, really good at-- It's still not a nice proof. It's not the [74:46] clean proof. It's not-- There's still a ton of work that needs to be done to make that proof a [74:50] Mathlib quality proof, and I think that's important work and work that we want to [74:54] do. But in terms of just getting that sorry-free theorem, they [74:58] just blasted through all of these. They jerry-rigged this, they jerry-rigged that, [75:02] and, you know, thirteen million lines later, they had a [75:06] sorry-free formal Lean proof of Fermat's Last Theorem. It's just absolutely [75:09] shocking. [75:10] - By the way, it's beautiful to always, um-- Because of how [75:14] rapid everything is, there's going to be a lot of statements that everybody makes, [75:18] the world experts make, that are wrong. That's the whole point of it. And that's the [75:22] fun thing about human discussion, is we have disagreements and I would [75:27] venture to say somewhere between forty to eighty percent of everything said [75:31] today will be wrong in five years because everything's changing s- [75:35] so fast. But that's the whole point of human discussion and deliberation and [75:39] discourse and the, even the scientific process is saying [75:43] shit and seeing what sticks, Over time. [75:47] Over ten years, over a hundred years, and so on. [75:49] - I was describing my experience. I had a lot of experience with Lean and, and [75:53] with using AI. I started doing Lean before AI existed in the, in [75:57] the sense that we understand it now, and it was very painstaking. And the second, [76:01] uh, AI systems came on, you know, the first LLMs were coming online, they were, they were crap at [76:05] Lean, but then they got a little bit better, a little bit better, and I was like, "Oh my God, I can do library [76:09] search." [76:09] I don't have to sit there looking, I know there's a lemma that does this, but I don't know what it's called, and I don't [76:13] know which exactly. Is it in the manifolds library? Is it in the [76:17] algebraic geometry library? Like, give me this, give me this lemmas. It was great at that, and I was [76:21] like, "Oh, this is amazing. I'm gonna keep using these systems." And every bit of my experience [76:25] told me that they can do, like, X, but not ten X, and this was a million X. [76:32] It was just completely outside of the distribution in my mind of what systems [76:36] were capable of. [76:37] - Do you have a ballpark estimate of how many humans were involved in that, [76:41] and, like, what is the level of human involvement in generating those thirteen million lines of [76:45] code? [76:46] - Uh, I believe it was two or three humans. [76:48] - Okay. [76:48] - And I believe most of their involvement was, "Keep going," all caps. [76:54] - Okay. All right, so you think it's really just, like gentle guidance, not like, [76:58] you know, pathway creating and- [77:01] - As far as I understand, there was no domain expertise- [77:04] - Okay [77:04] - ... on the parts of the people who were doing this. [77:08] - As the meme goes with OpenAI or Anthropic you know, the prompt is solve math- [77:13] ... or something. And then- [77:15] - Do a breakthrough. [77:16] - Yeah, yeah. Yeah, do a breakthrough. [77:17] - Do a breakthrough. That's the prompt. And it goes and does it. [77:21] - Of course, you never know from a company perspective, there's a lot of incentive [77:25] to sort of sell it as that, even though there might have been a lot of [77:30] intermediate steps that led whe- at any stages of the training or maybe some things [77:38] left unsaid, even though the final prompt is do a breakthrough, there could be a lot [77:41] of groundwork before that, that led to, we don't know. [77:45] - So I remember that was a Friday because I just was completely, [77:49] like my mind was completely exploded, and it was Labor Day, [77:53] and we just took the kids to the beach, and I needed a few days of, you know, I'm [77:57] not checking my phone, I'm not on Twitter. I'm just throwing, [78:01] you know, throwing my kids into the ocean and, and- [78:04] - Did you have that moment where you're just sitting and looking at the ocean and contemplating? [78:08] - Yeah. Like, what does it all mean? [78:11] So I had already gone through that a month before. [78:14] And I was like, I-- So now what that says is that [78:17] autoformalization is basically solved. [78:19] It's not solved in the way that I would want it to be solved. I want-- [78:23] And people are still gonna work very, very hard on this. [78:27] There's all kinds of other things that happen, just like there's all kinds of other things that happen when you prove [78:31] theorems that are byproducts of that process for human [78:34] understanding. There's all kinds of other things when you work on making a beautiful [78:38] library of mathematics, and that process will continue. That process will be [78:42] very difficult. It's gonna be the, you know, thousands of man-hours of people [78:46] around the world contributing in this open source way. But in terms [78:50] of can you get a sorry-free proof of something that human beings have written down [78:54] on paper, I think that problem is done. [78:58] - And so you were standing there with a whiskey, having gone through this a month before, [79:02] and now again. [79:03] - Right. This was round three. [79:05] - This is round three. [79:06] - 'Cause when I got back from Labor Day, it was Navier-Stokes. [79:08] - Navier-Stokes. [79:10] - But this one- [79:10] - So I was like, "I was just mourning FLT. You're gonna do this to me? Come on." [79:13] - But now, I mean, this really hit home, right? With, with the quasi-RH. [79:19] Um, so maybe can you speak to the, the importance and the beauty of the [79:23] primes, and also how big and hard the leap is from quasi-RH to full RH? [79:30] Yeah, so the primes, the reason the primes are so valuable is [79:34] that they are the fundamental particles in the theory of [79:38] multiplication. Every whole number is broken down into a product of [79:42] primes. You know, we know about the elementary particles [79:46] in physics, the, and all of matter is made up of that. In the same way, [79:50] the primes are the particles that make up the multiplicative structure of whole numbers. [79:54] And their distribution had been a mystery, and it remains a mystery. Even the Riemann hypothesis doesn't [79:58] solve everything we wanna know about prime numbers. [80:01] But the Riemann hypothesis was so beautiful because it pointed the way [80:05] to what the distribution should be, and it has a [80:08] thousand applications in very faraway fields. If [80:12] only the Riemann hypothesis were true, then we would know XYZ. [80:15] - So let me interject a little story that came up last week, which was [80:19] pretty entertaining. So we had Grant Sanderson was visiting IAS, and we had a [80:23] long discussion about what is AI doing to the math community. [80:27] And I really enjoyed this comment that Camillo De Lellis made, which was, [80:32] I'll do my best impression. I can't do the accent. [80:35] But there was a play written in the 1950s by a mathematician. [80:40] So who knows that mathematicians are writing plays, but apparently this [80:44] play, the premise of it was that Von Neumann died, and he went to heaven. [80:49] And he met God, and he got to ask him a question. So he asked him, [80:52] "Is the Riemann hypothesis true?" And of course, everyone holds their [80:56] breath as they're waiting to hear the answer. And God says, "Yes." [81:01] So finally, everyone can exhale. And then Von Neumann asks, [81:04] "But how do you know?" And God thinks and says [81:08] "Well, I know where all the zeros are, and they're all in the R equals a half [81:12] line." So my understanding is the premise of the play is that, [81:16] that's not enough for Von Neumann. He gets a little bit agitated to [81:20] try to find out more about why it's true, and I think he ends up at the end in the [81:24] other place. [81:26] - That's great. [81:27] - So I don't know what to draw as a moral from it, but, um- [81:30] ... it seems to speak to the mathematical experience. [81:33] - Math is not the fact, it's the why. [81:35] It's always been the why. It's our deepest quest for understanding. [81:39] And if you just tell me a fact is true, but I don't understand why it's true, that has [81:43] contributed zero to my knowledge. [81:46] - So primes, I think, are also interesting because they relate to [81:50] things that a three-year-old can understand. Like fundamentally, we know how to add [81:55] one to numbers, and you know, there's this breakthrough moment when you're [81:59] a kid, I can add one forever. That's a big idea. I can keep adding one forever. [82:05] Then you learn how to multiply and group, and that's also a big idea. [82:10] And primes are interesting to me, and have always been interesting, [82:14] because we don't understand how adding and multiplying work [82:18] together. The two most fundamental, basic, simple ideas in all of [82:21] mathematics, we don't know how they work together. I can add one to a number that I [82:25] can factor, and I have no idea what happens. I know if it's even, it becomes odd [82:31] and vice versa, but I don't know if a number that has many, many [82:34] composites becomes prime. We just fundamentally don't know [82:39] that simple, you know, adding one, how that changes multiplication. [82:43] And primes are basically, you know, this like, the [82:46] distillation of that three-year-old mystery, and it's a mystery that all of [82:50] us still wrestle with. [82:53] You know, that would be the interesting bit, is if the proof told me something [82:57] fundamental about that structure that I could actually use. [82:59] - Yeah. To the three-year-old, I think my older one was something like two or [83:03] three. He was playing-- We had these little blocks, you know, cubes that kids like to play with. [83:07] And uh, there was like 20 of them in the pack, and so he would make like a little four by [83:11] five grid, and he liked that, and he would make a 10 by two [83:16] grid, and that was kinda cool. And then one day, I look over and he's like really frustrated. [83:20] I'm like, "What's going on?" He's like, "You know, it's not, it's not working." [83:25] And one of them had rolled under the couch. And he only had 19. [83:28] And he couldn't figure out a way to make a nice-looking rectangle out of the 19. [83:32] - Yep. It still bothers me today. [83:34] - So... Yeah. So absolutely, kids love primes. Kids can- [83:38] ... discover primes. It's the non-re- [83:39] - And so the solution is to lose another, right? [83:42] - Yeah. Well, exactly. [83:43] - That's right. [83:43] - Yes. [83:44] - Do you think so the leap between quasi and RH, I [83:48] presume you think humans will not, in your lifetime, be able to [83:52] solve without AI's help, the full Riemann Hypothesis? [83:55] - I don't know. [83:56] - Can AI, I guess, is the follow-on question. Like, how hard is it? [83:59] - I don't know. I don't know. I did not think I would see a proof of quasi-RH. [84:04] Now that I see what the proof is, and there are a lot of technical details that I [84:08] still have to go through for my own satisfaction. There's a Lean formalization. [84:12] The Lean statement is, was already in the library, was already in Mathlib, so the [84:16] fact that they have a sorry-free proof gives me 100% confidence that the [84:20] proof is right. And moreover, having read the proof, you know, at a high level, [84:24] the ideas that it's using are all completely valid. They're brilliant. You know, [84:28] to embed the Möbius function, twist it by sextic [84:32] characters. This is all stuff that's the bread and butter of analytic number theorists [84:36] that was chained in just the right way and was absolutely incredible. And [84:41] so what Riemann showed is that remember we were talking about this [84:44] function that has complex inputs and has these outputs, and we're worried about [84:48] when the output is zero. So he showed that there's what's called the critical strip, [84:52] which is a vertical infinite strip in the complex plane, where the [84:56] real part is-- So you can think about the X coordinate lies between zero and [85:00] one, and the imaginary part, the Y coordinate, can go arbitrarily high or arbitrarily low. [85:04] And the hypothesis is that the zeros all lie right at the dead center of [85:08] that strip, where the real part is a half. That's exactly what Ankur was just [85:12] saying. And what we knew until Tuesday [85:16] was region where there are no zeros of the zeta function, [85:20] that's like getting thinner and thinner the higher up you go. It goes towards [85:24] the one line. The one line is something called the prime number theorem, which tells you the basic [85:28] distribution of the prime numbers is equivalent to the fact that there are no [85:31] zeros of this function on the line real part of S equals one. [85:36] So if the X coordinate is one- As you go up and down that line, you will never find a [85:40] value where when you apply the zeta function, it hits zero. [85:44] Okay? So we have this zero-free region that was getting thinner and thinner the [85:48] higher up you go, [85:50] and that gave us the best known distribution on the prime numbers, the best [85:54] error term in the prime number theorem, which I can go into if you-- You tell me how much math you [85:58] wanna talk about. [85:59] - Let's go. [86:00] - But what I could have anticipated is that it would [86:04] thicken that line a little bit as you, you know, it would still be going to [86:07] zero. The thickness would still be going to zero as you go up the [86:11] imaginary axis, but it would be better than what we currently know. [86:15] That thickness hasn't moved in seventy years, but I could sort of [86:19] imagine that it could maybe make some progress. What it did is it gave us a vertical [86:23] strip, so there are no zeros with real part between seven-eighths [86:27] and one, an entire infinite vertical strip. That's a quasi-Riemann hypothesis, [86:31] and if you can get that seven-eighths down to a half, that's the full Riemann hypothesis. [86:34] Now, the methods that it used in the seven-eighths proof, and there's a second proof that's about [86:38] eleven-twelfths, never mind these technical details, that method [86:43] sort of fundamentally can't get down to one half. [86:46] - So a new idea is needed. [86:47] - You need a new idea, but quasi-RH needed lots and lots of [86:51] new ideas, and it came up with that. [86:53] - Is your-- Having gone through this three times already, is your intuition, this is for [86:57] all three of you, about remeasuring in your mind the hardness [87:01] of the various problems? 'Cause you mentioned, like, picking easy things. [87:04] 'Cause that presumably you have a good hardness radar, like, this problem [87:11] is probably solvable, and this is not solvable today, [87:15] solvable this week, solvable this month, you know. Are you readjusting [87:20] to this new world and your- [87:21] - I'm readjusting my inability to predict it. [87:23] - Yes. Exactly. Well, you had the ability, no? It's noisy. [87:29] - Sure. It's a larger amount of noise than I thought. [87:32] I mean, I think we've always known that there are problems we care about that are within [87:37] reach of the tools that are out there right now, but it's just surprising the scale [87:42] at which, you know, how many different problems really fall into that category. [87:48] So we also don't know a lot about, you know, apparently, [87:52] they prompted thousands of problems, and what they dropped were only the [87:55] positive solutions, that they only had about a ten percent success rate.
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