Grant Sanderson Says Math AI Models Will Pivot Humans to Curators

Grant Sanderson：数学AI模型将促使人类转向策展人角色 人工智能在数学领域的飞速进展，正迫使研究重心从单纯的定理证明转向定义新概念、撰写通俗易懂的解释以及梳理跨学科联系。 Dwarkesh Patel 与 Grant Sanderson 探讨了人工智能如何重塑数学领域，涵盖从直接求解竞赛题到发现令人惊喜的跨学科关联。他们讨论了AI是否能构建全新的概念理论框架，或是生成人类易于理解的优美解释，而非不可解释的暴力证明。对话还探讨了自回归模型的局限性、“可反复试错磨练”（grindable）环境的重要性，以及随着AI的持续进步，人类数学家的角色将如何逐步转向策展与教学。 由于基准测试成绩突飞猛进，数学领域的进展已成为评估AI在其他领域能力的先行指标。 (0:00) AI通过暴力计算方法迅速解决了IMO几何题，但在需要创造性与谜题式思维的组合数学上却进展艰难。 (0:00) AI能够通过发现人类专家所遗漏的不同领域间的跨学科关联（“闪电”），从而加速数学发展。 (3:17) 与解决既定猜想或寻找反例相比，构建全新的概念性理论框架（“高山”）对AI而言是高得多的障碍。 (5:40) 对定义和猜想的评估依赖于学术共同体的定性认可，这使其难以直接适用标准的可验证奖励强化学习（RLVR）基准。 (9:15)

Grant Sanderson and Dwarkesh Patel explore how artificial intelligence's rapid strides in mathematics serve as a leading indicator for general intelligence, arguing that true capability will be measured by whether models can create unifying conceptual frameworks rather than simply brute-forcing solutions to pre-stated problems. Lightning Bolts, Mountain Building, and Spiky Frontiers (26:12) Sanderson describes progress in AI mathematics as having a 'spiky frontier' with an uneven, fractal distribution of difficulty. By 2024, AI systems could resolve International Mathematical Olympiad (IMO) geometry problems in under 19 seconds using structural brute force, yet they continued to struggle with the playful heuristics required for combinatorics. Looking toward grand challenges like the Riemann hypothesis, Sanderson categorizes potential AI breakthroughs into three archetypes: 'lightning bolts,' where an AI connects disparate existing fields that specialists rarely cross-reference; 'mountain building,' which requires creating entirely new theoretical machinery over decades or centuries, as seen in the proof of Fermat's Last Theorem using elliptic curves; and 'raw hustle,' which relies on sprawling, brute-force reasoning chains that leave humans with a tedious verification burden. The capacity to produce cross-disciplinary lightning bolts has historical precedent in human serendipity, such as when Hugh Montgomery and Freeman Dyson discovered that the zeros of the Riemann zeta function match the eigenvalues of random Hermitian matrices in nuclear physics. AI systems have already demonstrated this capacity in a narrow sense, such as discovering a counterexample to the unit distance graph conjecture and solving Erdős Problem 1196 by constructing conceptual bridges between previously separate areas. Century-Long Verification Loops and Mathematical Elegance (11:32) A fundamental difficulty in evaluating AI's capacity to formulate great definitions or conjectures is that scientific breakthroughs resist standard Reinforcement Learning with Verifiable Rewards (RLVR). Traditional benchmarks require immediate, verifiable signals, whereas the validation of a foundational concept can take generations. For example, Évariste Galois developed group theory in the 1830s to demonstrate why quintic polynomial equations cannot be solved using radicals. His papers were repeatedly rejected by contemporaries, and the broader utility of his framework took over a century to fully manifest in modern physics, such as Murray Gell-Mann's use of group theory to predict quarks in the 1960s. To bridge this evaluation gap, Sanderson proposes using compression metrics like Kolmogorov complexity to reward AI models for conceptual elegance. Patel notes that science prioritizes distilled, human-understandable abstractions—analogous to Newton's laws—over sprawling, uninterpretable computation. Rewarding models for succinctness could encourage them to generate unifying frameworks rather than thousand-page brute-force proofs. Grindability and the Formal Verification Bottleneck (52:19) Patel attributes the rapid pace of AI progress in code and mathematics relative to general web tasks to 'grindability'—the ability to containerize environments and run millions of deterministic rollouts in parallel without hitting rate limits or bot blockers. Containerization also neatly solves the credit assignment problem by linking precise code edits to measurable outcomes. While natural language reasoning currently powers major breakthroughs—DeepMind transitioned toward natural language at the IMO, and AI disproved the unit distance conjecture without formal proofs—relying exclusively on natural language creates a trust ceiling. As mathematician Alex Kontorovich noted, even a 1% error rate in AI-generated papers would overwhelm human peer review. Formal verification systems like Lean provide an automated 'green checkmark' of absolute correctness. Sanderson observes that this opens up unsupervised mathematical exploration, allowing automated systems to continuously expand libraries like Mathlib or search millions of arbitrary axiom systems, as explored by Terence Tao, without requiring human oversight to validate basic correctness. Human Curation, Exposition, and the Economic Test (34:14) As AI models automate theorem-proving, Sanderson anticipates that human mathematicians will shift toward becoming conceptual 'curators' and expositors who select which problems are worth pursuing. Referring to Timothy Chow's concept of 'unsolved expository problems,' Sanderson distinguishes a formal, step-by-step proof from a genuine explanation of why a statement is true. While LLMs excel at distilling known concepts, their lack of physical embodiment and theory of mind makes generating unpredictably insightful prose difficult. Sanderson views teaching as a resilient human role in an automated future due to the inherently social and relational nature of mentorship. Economically, the impact of accelerated mathematics depends on practical spillovers. Theoretical advances in partial differential equations already yield massive industrial savings—such as allowing Boeing to replace expensive physical aircraft teardowns with computational fluid dynamics simulations. However, Sanderson and Patel caution that if a tenfold acceleration in AI math fails to produce tangible real-world applications beyond solving abstract contest problems, it may uncomfortably expose modern pure mathematics as increasingly decoupled from physical reality. The conversation shifts the benchmark for AI progress from mechanical problem-solving to the harder task of conceptual curation: formulating fruitful definitions and deciding what questions matter. Whether advanced AI will autonomously generate elegant, practically relevant theories, or remain an engine of brute-force deduction requiring human interpretation, remains an open question.

3Blue1Brown 创作者 Grant Sanderson 与 Dwarkesh Patel 指出，数学作为人工智能能力演进的风向标，其突破正由解答既定难题转向提出底层定义与跨学科框架。随着模型在定理证明与反例构造上展现惊人进展，人类数学家的核心角色将从单纯的解题者过渡为把握研究方向、评估概念美学与提炼直觉的“策展人”。 数学能力的尖刺边界与基准漂移 (0:00) Sanderson 将当前人工智能在数学领域的进展特征描述为具有分形结构的“尖刺前沿”（spiky frontier）。到 2024 年，AI 系统已能够通过高速的结构化暴力搜索方法，在 19 秒内基本“彻底解决”国际数学奥林匹克（IMO）的几何题，但在更具跳跃性与游戏感的组合数学上仍然表现挣扎。这种极端的不平衡性证明，通过特定竞赛基准并不等同于实现通用人工智能（AGI）。 随着 AI 开始在单位距离问题等领域通过构造反例推翻著名猜想，评估指标的目标线随之迁移。传统基于可验证奖励的强化学习（RLVR）高度依赖确定的对错判定，但这无法评估“提出伟大猜想”或“创造全新定义”的能力。Sanderson 观察到，在 2025 年中至 2026 年间，数学研究者的态度发生了显著转变，他们开始在日常对话中利用 AI 帮助判断哪些初始问题真正值得被深入探索。 闪电跨界与漫长验证周期的理论构建 (3:17) Sanderson 将数学突破的路径划分为两类：一类是横跨不同领域的“闪电”联结，例如当年物理学家 Dyson 在普林斯顿高等研究院听到数论学者 Montgomery 描述黎曼猜想零点分布公式时，迅速辨认出其与量子物理中随机矩阵特征值的数学同构，而具备海量知识广度的人工智能有望系统化地复现这种跨学科洞见；另一类则是费马大定理式的“建山”过程，需要历经数个世纪打造如椭圆曲线与模形式等庞大理论框架。 科学中最深远的突破往往伴随着跨越数十乃至上百年的超长验证周期。从拉格朗日研究根的置换性质，到伽罗瓦在狱中萌发群论雏形以证明五次方程没有通用的根式解，再到盖尔曼在 20 世纪 60 年代运用群论预测夸克，其理论力量耗时一个多世纪才彻底释放。这种漫长的回馈周期使得单纯依赖即时验证奖励来训练 AI 的“数学直觉”面临巨大的方法论挑战。 可打磨性差距与形式化验证的意义 (52:19) 讨论分析了为何 AI 在数学与代码上的进步速度远超一般性现实环境任务。Dwarkesh 认为，真正的瓶颈在于系统的“可打磨性”（grindability）与归因清晰度。代码和形式化逻辑可以在容器中以极高速度进行数万次并行的确定性推演，能够直接将结果与精确修改绑定；而网页操作与商业环境受到限流、防抓取和不可控变异的阻碍，无法高效生成合成经验。 尽管 AI 在解决 Erdős 1196 题或破解单位距离猜想时主要采用自然语言思维链而非 Lean 等形式化语言，但形式化工具对于长期探索不可或缺。随着 AI 论文吞吐量激增，只要存在 1% 的潜在逻辑漏洞，人类就难以承受审阅负担。Lean 等证明助手提供的“绝对绿色对勾”构成了独一无二的自动验证底线，同时也使不受人类干预、自主扩充数学库（Mathlib）的无限探索成为可能。 证明自动化下的策展人与阐释者角色 (29:55) 当证明的机械生成被大规模自动化，数学界将直面 Timothy Chow 所称的“未解决阐释问题”——即命题已被严格证明，但学界仍未理解其背后为何成立的根本机理。Sanderson 强调，严格的形式证明与深刻的概念解释截然不同，真正的理论先驱往往也是极具洞察力的解释者，就像闵可夫斯基时空图不仅用于展示相对论效应，它本身就构成了对物理实在的核心洞见。 未来人类数学家与教育者的职能将向“艺术馆策展人”演变。由于纯文本自回归模型缺乏身体微模拟等具身硬件和对读者心智状态（Theory of Mind）的精准预判，AI 在撰写需要打破预测惯性的卓越文章上依然受限。在 AI 探索生成的无垠数学理论中，人类将负责鉴别哪些概念方向具有审美和实用价值，并承担起具有深层社会连接属性的教学与导师职责。 纯数学加速与工程现实的潜在脱节 (89:07) 若人工智能推动纯数学研究取得十倍或百倍的加速，这种智力产出能否直接转化为经济动能仍存疑问。Sanderson 指出，偏微分方程（PDE）和动力系统等细分领域的理论推进可以直接大幅优化计算流体力学（CFD）模拟，例如帮助波音公司完全通过计算机模拟替代成本高昂的实体飞机拆装测试；但代数数论等高度抽象领域在短期内极难与物理现实产生直接互动。 讨论最后指出，如果 AI 驱动的数学大爆发最终仅限于攻克一系列孤立的 Erdős 竞赛型难题，却未能在物理设计、材料科学或更广泛的工程系统中引发外溢，可能会让纯数学界长期以来以“未来衍生应用”为由争取研究经费的传统论调面临严峻审视与尴尬反思。 该讨论的核心未解问题在于：当人工智能在解决预设数学难题上超越人类后，缺乏即时验证反馈的定性判断——例如挑选何种公理体系更具美感、何种定义更具长远价值——能否被形式化或被算法内化。如果纯数学的爆发式增长无法向物理现实和现实工程有效外溢，数学研究与人类社会经济需求之间的脱节或将被进一步放大。

[9:15] Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions [34:14] Grant Sanderson: Set a man on fire, and he's warm for the rest of his life. [38:08] Grant Sanderson: Imagine I've locked you in a box, and the only way you have of interacting with the world is that you receive a slip of paper, and someone says, 'Can you predict what will come next?' You predict what will come next, and then your memory's wiped. You get another slip of paper. [47:47] Albert Einstein: God does not play dice [52:23] Andrej Karpathy: sucking supervision through a straw [56:48] Grant Sanderson: What stands to be interesting... is that with Lean, you could imagine having a basically endlessly running program that's constantly trying to extend Mathlib... It doesn't need anybody to check in on it. It could just keep going. [67:07] Dwarkesh Patel: When you're writing, you sort of know that in order for it to be good, you have to have an element of the unpredictable... It's knowing exactly the correct point when you want to make an unpredictable move, and that that's going to be what's more insightful. [92:25] Grant Sanderson: Ugh.

[0:00] Today, I'm chatting with Grant Sanderson, who runs  3Blue1Brown and is now working on a new project [0:04] documenting the progress AI is making in math. I wanted to talk to you about this because AI [0:09] has been making the fastest progress  in mathematics out of any other field. [0:13] Whatever is happening here, and whatever way  we're seeing AI progress happen or not happen, [0:17] will tell us about what will happen to the  rest of the world as AI gets better and better. [0:21] I wanted to start with this question I asked you  when I first interviewed you three years ago. [0:26] I asked you, once we have AIs that can get  gold in the International Math Olympiad, [0:30] wouldn't that just be AGI? Wouldn't this just be able [0:33] to do anything any human can do,  given how hard these problems are? [0:35] You had an answer, which in retrospect  turned out to be very wise and correct. [0:40] You said it'll be another benchmark, like all  these other benchmarks that AI are passing. [0:43] Obviously, AI has gotten better in  a general way since then, but there [0:47] won't be some "aha" moment when this happens. First, I'd be curious to get your heuristics [0:54] on why that turned out to be true. Second, I'm curious how long you [0:58] think this narrowness can continue to be true. By the point that AI has solved a Millennium [1:03] Prize problem, do you think it's still possible  that there are lots of tasks humans are doing [1:09] that AI still can’t automate in the economy? It's an interesting question because it's hard [1:13] to answer without knowing what the  solution looks like ahead of time. [1:16] If we take the IMO, the spirit of  your question three years ago was in [1:22] looking at how some of the solutions to these  problems really seem to require creativity. [1:26] The designers of these problems try to come up  with things that you can't train for as easily. [1:32] The dirty secret with the IMO is that  you really can train for a lot of them. [1:38] With the whole AI and math project underway, as  you point out, one of the reasons it's interesting [1:43] at all is that there's a spiky frontier to AI,  and math is just right there in one of the spikes. [1:48] But there's a fractal nature to that  spikiness, because when you zoom into [1:51] the specific progress within math, you have  some things that are a lot easier than others. [1:56] If we just think about IMO,  which is old news at this point. [1:58] It's been two years since  they're really doing quite well. [2:01] They would have gotten a gold in  2024 if not for the following reason. [2:05] They're very good. They just  cold-solved geometry basically. [2:10] The IMO has these four  categories of problems: geometry, [2:13] number theory, algebra, and combinatorics. Geometry, it just solves it in nineteen seconds [2:19] since 2024 because it's a brute force solver. The dirty secret is that for students, there's [2:24] also a brute force way you can go at it. Combinatorics is the wild card: [2:29] much more playful, puzzly-seeming problems. There were two combinatorics problems on [2:35] that year's test, and there's not always. There are four categories and six different [2:39] problems, so it's a toss-up which  one is going to have two questions. [2:43] Had it been more geometry questions,  they would have gotten a gold that year. [2:47] But it struggles on those combinatorics ones. Someone who's trying to keep that torch of [2:52] the last holdout of math for humanity might say  those are the ones that require more creativity. [2:59] Even then, the spirit of your question—if they're  solving a Millennium Prize problem, does that also [3:06] service a lot of white-collar work?—suggests that  whatever the rate limiter is between where we are [3:12] now and that is the same as the rate limiter  for making things better at white-collar work. [3:17] We could paint a couple of different ways. If we focus on the Riemann hypothesis, [3:21] what would it look like to solve that? These things are extremely good at a specific [3:30] domain of knowledge, knowing it very deeply,  and then knowing another domain, and another. [3:35] You've pointed this out. It's bizarre to  have something with this superhuman breadth [3:39] that knows all the fields so well, and yet isn't  finding those lightning bolts that connect them. [3:44] I think we're starting to see sparks of  it actually finding connections between [3:48] the things it's an expert at. I'm sure we'll talk about it. [3:51] If the nature of the solution to the  Riemann hypothesis was something like that, [3:55] that feels pretty distinct to me from what's  necessary to get good at white-collar work. [4:00] And there's a reason to believe that  might be the nature of the solution. [4:04] I don't know if you know the story of Hugh  Montgomery and Freeman Dyson at the IAS. [4:11] This is a side tangent, but it's a fun story. I don't know if it was over lunch or something [4:16] like that, but you have this number  theorist who is just trying to understand [4:21] the statistical correlation between pairs  of zeros of the Riemann zeta function. [4:25] The Riemann hypothesis is all about whether  all these zeros sit on a straight line. [4:29] He finds this quantitative question you  could ask, and he writes down a formula. [4:32] It looks like one over sine  squared or something like that. [4:35] Freeman Dyson, a physicist, is  like, "I know that expression. [4:38] That expression comes up in studying  the eigenvalues for random Hermitian [4:42] matrices," which was something that comes up  in studying the energy levels of a nucleus. [4:48] The idea that the statistics of those two  seemingly different things were the same [4:52] prompted an exploration of whether there  are aspects of random matrix theory that [4:58] might be relevant to the Riemann zeta function. I think it's a little bit of an open question [5:02] whether there is fruit to be had there. But that bridging together of two different [5:06] fields—if it turned out that the solution to  the Riemann hypothesis was exploring an idea [5:12] like that even further—has the character  of how you expect LLMs to be good at math. [5:17] They're experts at quantum physics. They're experts at analytic number theory. [5:20] They should be able to see that similarity in  a way that doesn't require Montgomery and Dyson [5:25] to be having lunch and happen to talk about it. That's totally different from white-collar work. [5:30] To the extent that you have a hard  time using an AI as an editor, [5:36] it's not because they know everything and you just  need them to find that lightning bolt in between. [5:40] A different possibility would  be… What's the right analogy? [5:46] Maybe if we think of Fermat's Last Theorem,  between the moment of Fermat phrasing the [5:49] question and what the solution itself looks  like, where the solution ultimately involves [5:54] such heavy machinery in math. The beauty of that problem is [5:58] you can phrase it so simply. You ask about xⁿ + yⁿ = zⁿ. [6:03] Do you have integer solutions for  this when n is bigger than three? [6:09] It's something you might expect there to  be an elementary number theory approach to, [6:12] but as far as we can tell, there's just not. Whereas the actual solution, maybe there is [6:17] something simpler, but this  might be what it has to be. [6:22] There’s such a complicated set of  ideas that build on centuries of work [6:27] centered around elliptic curves. Then there's this other mountain of ideas [6:30] centered around these things called modular forms. Both of those mountains have to be built before [6:35] you can ask the right question that connects them. If the solution to the Riemann hypothesis involved [6:39] building a new mountain, that's a kind of  skill—the ability to come up with the right new [6:44] ideas—that feels sufficiently different from the  character of how they're intelligent right now. [6:51] It's not like that's what you need  from your hired video editor per se. [6:54] But if it's capable of building mountains that  are the correct new theory crystallizing how we [7:01] should be thinking about a subject, that's  just such a level of intelligence that it [7:07] would be surprising if it didn't permeate  into other aspects of the economy besides [7:11] just the mountain-building for math itself. Or at the very least, even if it couldn't [7:15] literally do every single thing white-collar  humans can do, it would just have transformative [7:20] effects in the way that getting gold in the IMO  did not have transformative effects on the world. [7:24] First of all, I do want to point out that  I'm totally moving the goalpost here. [7:29] When I interviewed Dario two or three  years ago, I asked this question about [7:31] why they haven't been able to use their  vast knowledge to connect ideas together [7:35] and come up with a new discovery that way. That seems like the kind of thing where even [7:38] a moderately intelligent person,  if they knew this much information, [7:40] would be able to come up with a medical diagnosis  from the fact that this drug causes migraines, [7:45] and this other thing does this, and maybe  it's the same drug that can cure both things. [7:52] From an outsider's perspective, mathematics  seems clearly like a field where finding the [7:56] counterexample to the unit distance problem  conjecture was an example of this kind of thing. [8:01] So it’s total goalpost moving. But then we can ask, what is the next benchmark? [8:05] Now that AIs can do this thing we should  have thought they'd be able to do, what is [8:10] the next thing that would be quite impressive? There are a couple of candidate ideas here. [8:15] One could be coming up with interesting  problems in the first place, and the [8:18] other is coming up with new kinds of objects or  conceptualizations that create or unify fields. [8:25] On the first one, right now we have  these Millennium Prize problems [8:31] because mathematicians have noted them. Riemann came up with this idea of the [8:36] Riemann zeta function because he thought  the zeros of this function would have [8:42] some connection to the density of prime numbers. Figuring out why we think this is an interesting [8:47] thing to study in the first place, why  we are building this object and trying [8:51] to answer questions about it—and answer this  particular question about it—seems like the [8:55] kind of thing that would be the next benchmark. You highlight two pretty good examples there. [9:03] For anyone curious about the unit  distance conjecture, there's this [9:06] really nice video by a math channel  called Polylog where they talk about it. [9:11] All of these discussions cause people  to reflect on the process of doing math. [9:15] They're like, "Oh, this thing  can do this impressive stuff. [9:17] What does that mean for us?" One of the people in that video highlights this [9:20] quote: "good mathematicians prove theorems, great  mathematicians come up with conjectures, and the [9:26] greatest mathematicians come up with definitions." That's more or less exactly your framing here. [9:31] We need the conjecture generator  and then the definition generator. [9:35] That's the premium-tier  mathematician. I don't understand [9:39] how exactly you'd make that a benchmark. Usually, when I think of the word benchmark, [9:44] I'm thinking of something that is a goalpost. The ball is through the goal or it's not. [9:48] You can clearly say, "Yes, this is done." Partly that's to be able to do things like RLVR, [9:53] but also partly just to know that you  haven't moved the goalpost in answering. [9:58] OpenAI can have their headline  on disproving the unit distance [10:01] conjecture because it's a clear, distinct thing. It did it. Whereas imagine trying to have a [10:06] headline on GPT-5.4 coming up with a  really good conjecture. "We promise, [10:11] everyone thinks it's a good conjecture." It just doesn't land the same way. [10:16] But maybe that doesn't negate the fact that  it's the right thing to be thinking about. [10:21] I would be surprised if it ever took  the form of looking like a benchmark, [10:24] where we have a score saying it's passed because  we can quantify how good a conjecture is. [10:30] The nature of what it would take is  probably that you'd feel a tone shift [10:34] in conversations with mathematicians  about the way it's useful to work with. [10:40] This series you referenced, which is not  at all produced yet and probably won't be [10:43] for a couple of months, takes the form of  us interviewing a lot of mathematicians. [10:48] What's interesting is that we started doing this  over a year ago, and it's fun to see a little bit [10:52] of a tone shift in the way they talk about AI  between mid-2025 and where we are now in 2026. [10:59] In the real world, that's a  very short amount of time. [11:01] In the AI world, that's eons. We're able to see this tone shift over those eons. [11:06] I think the way you'd measure  conjecture-generating ability is going [11:10] to be more subjective, based on that tone shift. It will be mathematicians saying they're not just [11:15] using it to solve their problems, but that as they  step back and decide what their research field [11:19] should even be, a conversation with such-and-such  model was genuinely helpful for that. [11:24] I don't think it's likely you'd see  it in the form of a headline saying [11:29] this was yet another benchmark knocked down. It's very interesting. The kinds of things [11:34] you can't make benchmarks for are also the  kinds of things, at least in the current [11:38] paradigm, you can't easily train for. There's really no fundamental difference [11:42] between a benchmark and a training environment. It's very easy to come up with some dichotomy of, [11:50] "here's a deep reason why AI can't do  a certain thing", and then it turns out [11:53] you're just thinking about it the wrong way, and  actually it can do it pretty soon thereafter. [11:57] But I'm going to come up with— You're going to come up with a couple anyway. [12:04] It'll probably turn out that there are  ways we can train AIs to do these kinds [12:07] of things in the relatively near term. But it seems like it would have to be [12:10] different from current RLVR training. The thing I'm curious about—and the [12:13] thing that seems to me to drive a lot of the  big progress in mathematics and in science [12:18] generally—is coming up with a new way to think  about a problem or a new way to understand the [12:25] world that unifies different fields, spawns  entire new fields, and solves problems we [12:31] weren't even trying to solve in the first place. The reason Einstein was thinking about GR is [12:38] not because he wanted to explain why  light bends or why black holes exist. [12:41] These are phenomena he didn't even  need explained in the first place. [12:45] In mathematics, as a total outsider who  doesn’t even know what he’s talking about here, [12:51] it seems like there are often ways to prove  a specific problem that can motivate a new [12:56] conceptualization—one which results in a whole  new field, a whole new way of thinking, which is [13:02] immensely productive—and ways which don't. I'd be curious to hear you talk about [13:08] Galois coming up with group theory, distinguishing  his solution to the quintic having no formula [13:13] for the roots, and Abel coming up with a  different proof a few years earlier that [13:17] didn't come up with group theory. If you wanted to do a verification [13:19] loop on whether group theory is an interesting  concept—was something useful done here, or why [13:24] is this proof better?—potentially that  verification loop is a hundred years long. [13:28] It involves cryptography coming  around and physics making progress, [13:32] and the ideas in group theory being relevant  to understanding symmetries in physics. [13:37] There's a hundred-year verification loop on why  this is a productive concept in the first place. [13:44] You struck a nerve, because I had this project  about Galois I was going to do in 2022 that I [13:49] put on the shelf, but I spent a year of  my life thinking a lot about what he did. [13:53] There's a risk of me accidentally talking too long  on the specifics, which you can hold me back on. [13:59] It's a perfect example for your case, because  describing why it was a valuable insight [14:07] does not come from immediate utility. Certainly, if you're thinking about [14:11] RLVR environments, this is  going to be really hard to do. [14:15] But it's interesting to note that even with human  verifiers at the time, it took a really long time [14:20] to recognize it as being useful. With Einstein and GR, people could [14:25] feel this was a good theory right away. What makes Galois theory such an interesting [14:29] example is that you literally have this  hundred-year segment of an idea that flows through [14:35] many different people's heads before it settles  into something the math community agrees is good. [14:41] To back up a little bit… Do you want  the background on the problem at all? [14:48] We all learn about the  quadratic formula in school. [14:51] I thought you were going to say we all learn about  group theory in school, but I missed that class. [14:54] We all learn about group theory…  No, the quadratic formula. [14:58] This was known. In some sense,  the Greeks could solve quadratics, [15:01] but they didn't really write things in algebra. It's really the Arabs who wrote down that formula. [15:07] There's this delightful story about dueling  Italian mathematicians—not real duels, just [15:14] intellectual challenges—who secretly  found a formula for the cubic, [15:20] and then very shortly thereafter found  a formula for degree-four polynomials. [15:24] So a natural open question for mathematicians is, [15:27] can you find a formula that  solves degree-five equations? [15:30] The degree-four formula is a monster. It would be wild to write it down. [15:35] You usually don't write it down in full. You break it up as a procedural thing. [15:39] You might believe these things have this  exponentially increasing complexity. [15:42] So for many hundreds of years, nobody  was really answering that question. [15:45] Usually, we say Abel was the first to prove it. He was this young, [15:50] precocious Norwegian mathematician. He showed it's simply impossible. [15:54] It’s not that you can find a quintic formula. He thought he found one initially, but [15:56] he showed it's impossible. I think the real credit though, [15:59] you have to back up a bit and talk about Lagrange. He found the right kind of [16:05] question to ask about this. I'll give it at a very high level. [16:13] He was studying the question and recognized  that being able to solve these polynomials [16:18] is very related to understanding the way  certain algebraic expressions are symmetric. [16:24] If I write down a + b + c + d, just adding  four variables, and I permute those, [16:29] it doesn't change the value of the expression. Whereas if I write a + b * c + d, some of the [16:35] permutations don't change it, but some of them do. He had this really nice insight about how if you [16:40] can find expressions that have four free  variables, but all the permutations take [16:44] on three distinct values, that has this  unexpected relationship with being able to [16:47] reduce degree four into degree three. He started approaching the question [16:51] of whether we can find a quintic polynomial  by wondering if he could extend that method. [16:56] To extend that method, you would have  to have an expression that has five free [17:00] variables such that as you permute them  over all the five factorial permutations, [17:04] it takes on only four values or fewer. You could put that in a puzzle book. [17:08] You could put that in a brain teaser  that a twelve-year-old could engage with. [17:13] It's not too hard to find yourself  feeling like that's an impossible task. [17:17] Lagrange is sitting there saying,  "Here is a strategy to solve this [17:21] problem of finding a quintic polynomial. It seems like it might be impossible, [17:26] at least from this strategy." But that was the first time in [17:29] history that people had the instinct that some  kind of question about symmetry was the right [17:34] way to study these polynomials. In his mind, it was just a way. [17:37] It had yet to be discovered that there  was actually a tighter connection. [17:40] Also maybe rather than searching for the formula,  we should be asking the opposite question: [17:44] can you prove that it's impossible? He sort of planted that seed. [17:48] Around fifty years later, Abel definitely  read Lagrange and was influenced by it. [17:53] We know that Galois loved Lagrange  when he was falling in love with math. [17:56] It's very hard to imagine that these two young  geniuses coming up with pretty similar insights [18:02] around that problem wasn't born from Lagrange. But to your question on whether you are able [18:07] to verify that this was a good idea, there  wasn't any result that Lagrange came to. [18:13] He didn't solve the problem, so it  wasn't a case of knowing it was the [18:16] right question to ask based on a solution. He just asked it. There's some intrinsically [18:19] interesting thing about it. It also wasn't very [18:21] important for math at the time. Most people were more interested in [18:24] the applications to physics. This was almost a side, [18:28] recreational, hobbyist-type thing. Abel started working on quintic stuff, [18:32] but then he was advised to spend more of his  efforts studying elliptic functions, so more of [18:38] his work was on that before he died young. He died at twenty-six from tuberculosis. [18:42] And then Galois pushed both of those ideas in  the right direction, where he really understood [18:51] the nature of abstraction. He had this really nice piece [18:54] that he wrote while he was in prison. We could talk all about his life story. [18:58] It's pretty wild. But he's this teenager,  he's in prison, and he had tried to submit [19:02] his math papers and they had been rejected. So again, thinking about verifiable reward, [19:06] the verifier function that is the academy  at that time is rejecting what he wrote. [19:11] Frankly, it was not very coherent. It wasn't a complete proof. [19:13] He wasn't giving a clear thought  of what the theory actually was. [19:16] He was just a young fledgling  mathematician getting his bearings. [19:20] The verified reward there is, "No good." But he has some instinct that there's [19:24] something there. So he's writing this [19:26] diatribe on the nature of math being something  that undergoes these shifts over time. [19:33] He talks about the advent of algebra itself  and going from just thinking in terms of [19:39] numbers to having a certain fluency with pure  algebraic expressions, where you're not tied [19:43] to interpreting those expressions. He has this instinct that there [19:47] seems to be another layer of abstraction that  we should be doing, where rather than thinking [19:51] about the formulas themselves, we're thinking  about what symmetries underlie those formulas. [19:56] But it was still a pretty ill-defined theory. If you're trying to say the verified [20:00] reward is that he solved a problem that other  people haven't, well, Abel already proved that [20:04] quintics are unsolvable. So what was Galois doing? [20:08] In principle, Galois theory lets you  take a specific polynomial, and it [20:12] gives you the rules to say whether that specific  polynomial has roots that you could write down. [20:16] For example, with x⁵ - 1, you  know that a solution is 1. [20:20] Or x⁵ - 2, you can write  down the fifth root of two. [20:23] So it's not that you can't write down the  solution for every quintic polynomial, but [20:26] could you find a specific one where you prove  you can't write the solution using radicals? [20:30] He also didn't even solve that exactly. He didn't show for a specific example [20:36] that he couldn't. Even describing [20:37] what problem he solved is very tricky. He then dies. It's this very romantic [20:42] story of him having this duel. There's a lot of myth around how [20:47] he supposedly writes up all his ideas the  night before the duel, but really, he tried [20:50] to get them published five times before. Working on the quintic doesn't seem to [20:51] be good for your health. It's very bad. If you're [20:54] a young genius, don't work on the quintic. He asks his brother and his close friend to get [20:59] his notes to Gauss, to get these notes to  the important mathematicians of the day, [21:03] because he thinks there's something there. Even then, it didn't really take. [21:07] His brother and his friend tried to get them  out, but it was another twenty years until [21:11] Liouville sees these notes, sees that maybe  there's something in them, and tries to clean [21:15] them up and understand what Galois was getting at. Even then, it was another twenty years or so until [21:21] Jordan actually puts together something  like a modern treatment of group [21:27] theory that they attributed to Galois. You could easily imagine history turning [21:32] differently, where these ideas were  coming about from other points in math, [21:35] and Galois could have been forgotten in  history if he was a less florid character. [21:39] But between the time of Lagrange having this  inkling that maybe symmetries of roots is [21:45] the right way to go, to where it all looks like  modern group theory, you've got this long span. [21:50] A lot of the time, it's not even passing  the verified reward of human reviewers. [21:54] It gets on someone's desk and they say, "I  don't really know if there's anything here." [21:58] You have to have this one person recognize it. Even then, it's not really solving practical [22:04] problems at that point. You pointed out cryptography [22:06] and physics and things like that. You have to get into the twentieth century [22:09] before you have Gell-Mann thinking that  maybe understanding the nature of how certain [22:16] groups break down has a relationship  with what particles are made out of. [22:21] He anticipates quarks based on a  purely group-theoretic question. [22:25] That's one of the more interesting applications  of group theory: to even predict the existence [22:30] of quarks is a group-theoretic question. That's so long after Lagrange before you [22:35] have anything like that. So you have to ask, [22:39] what is the way of measuring progress that's not  based on solving a problem, but that is somehow [22:46] capturing the instinct inside Galois's mind  when he says, "I think there's something here"? [22:50] What's the instinct inside  Lagrange's mind when he says, [22:53] "I think this is the right way to think about it"? What's the instinct inside Liouville's mind [22:56] when he says, "These scattered notes from this  long-dead youngster might have something to them"? [23:01] It's so hard to put a finger on that. A different series of videos I'm [23:07] making right now is about the whole  "compression is intelligence" idea. [23:12] Even though this isn't really the angle  I'm taking, there is something to the [23:15] idea that the smaller expression that's  more predictive feels more intelligent. [23:22] So I wonder the extent to which you can give some  kind of verifiable reward around not just whether [23:27] you solved it or what it is solving, but around  the smallness of the concepts required to do it. [23:34] Going back to Riemann hypothesis solutions,  what would that look like if an AI solves it? [23:37] I think a third way it could happen  is it just straight-up works harder. [23:41] In the same way, you could maybe have an  elementary proof of Fermat's Last Theorem [23:44] that's just spelled out over thousands  of pages that would be incoherent. [23:48] But the cleaner way to view it is  with elliptic curves and all that. [23:51] Maybe there's some thousand-page proof of  the Riemann hypothesis that no one's really [23:56] getting anything out of, and what  you actually want are the succinct, [23:59] compressed versions of those ideas that would  then lend themselves to human understanding. [24:07] Maybe you throw Kolmogorov complexity into your  attempt to quantify what you mean by elegance. [24:14] I don't think it's easy, but I do think it's  something you would have to do in order to [24:18] reward the Galois-like instinct, rather than  just rewarding whether you solved a problem. [24:23] It's very hard to come up with  the heuristic for science. [24:27] But it's clear humans have been  doing this somehow, and obviously, [24:31] AIs will do it at some point. It's relevant also not just in [24:34] terms of verified reward, but presumably, the  end goal is understanding, human understanding. [24:40] Even if you do have some thousand-page  proof of some math thing or some grand [24:44] new physical theory, the goal is understanding. Maybe if the goal is predictiveness, you can [24:49] just have automated engineers go off and build  rocket ships where we have no idea how they work, [24:54] but we can get between stars. But there are going to be a [24:57] lot of people who want to understand. You're still going to want whatever the [25:01] concision function is that distills down this  complicated way of thinking into the right one, [25:06] like the equivalent of the universal  law of gravitation for Newton. [25:10] You would still want to train AIs to be able to  do that and find the compressed representation. [25:16] I grew up in India until I was eight, so in  addition to English, I also speak Gujarati. [25:21] And since Google just released Gemini 3.5  Live Translate, I thought it'd be fun to [25:25] put it to the test in this mid-roll. (Gujarati) Gemini 3.5 Live Translate [25:31] automatically detects more than 70 different  languages and translates in almost real-time [25:36] into the target language. Live translate your original [25:40] speed and format while speaking. Just like it’s doing right now. [25:45] I visited China back in 2024, and I remember  thinking at the time that the trip would've [25:49] been so much more productive if I'd been able to  live-translate the conversations I was having with [25:54] researchers and random people I met on the street. Now we have that technology. [25:57] So if you're building an app that needs  live translation, you should 100% check [26:01] out Gemini 3.5 Live Translate. It's available now via the [26:04] Gemini Live API and in AI Studio. Go to ai.studio/live to get started. [26:12] People have this worry about mathematics  in particular that AIs will prove the [26:16] Riemann hypothesis, and our understanding of  mathematics won't be any the better for it. [26:21] I have a couple of questions about this. The first one is whether this is [26:24] something you should expect. Isn't the reason humans come up [26:29] with general, natural objects and subgoals  when we're working on a big problem that [26:36] this is just useful when you're trying to  work on a complicated, important problem? [26:42] Theoretically, would this even be a simpler way to  solve the Riemann hypothesis, as opposed to just [26:46] coming up with the natural abstractions that  are relevant to thinking about the problem? [26:50] And then two, empirically, is this what we  observe when AIs make progress on problems today? [26:54] When the AI came up with that counterexample  to the unit distance conjecture, [27:00] you can just read its chain of thought. It's not understandable to me, because I don't [27:04] know anything about mathematics, but it seems that  to other mathematicians it was understandable. [27:09] It made use of known concepts of mathematics  and proved relationships between them, [27:12] all in natural language. As a result, it accelerated [27:15] our understanding of the connection  between this object and this conjecture. [27:22] Empirically, is this a thing  we should be worried about? [27:24] I think it depends on the nature… If we break  down the three possible ways of solving the [27:30] Riemann hypothesis… The other big one from this  year was a certain Erdős problem numbered 1196, [27:38] about these things called primitive sets. It had that character of bringing an [27:43] idea from a seemingly different field. As soon as you present the basic idea to a [27:47] mathematician… You say, "What if we try the Markov  chain process where we show that this thing is one [27:54] from the bottom up probabilistically rather than  the top down, and use the von Mangoldt function?" [27:59] If you say that to someone in the  know, they’d know how to run with it. [28:02] You have this very small idea that has the form of  expertise in one field and expertise in another, [28:08] drawing a little lightning bolt between them. Those are going to be very human-parsable, [28:13] because all you have to do is show the start  and end point of what those connections are. [28:17] If the character of it is mountain building, you  have to put in a lot more time to understand that [28:22] new mountain that was built, because it's a new  thread, not just a lightning bolt between them. [28:26] And if the nature of the progress was just  raw hustle—a super long chain of reasoning [28:35] with no new theories—then you would have  that worry of this whole digestion process. [28:38] So I don't think there's one clear answer. It depends on what the solution would look like. [28:44] On the mountain building side, that would  actually be really interesting to see. [28:48] Is it by default very human-understandable, the  way we see new theories from great mathematicians? [28:54] Or is it an alien, different kind of mountain  being built where we have to reprocess the [29:00] kinds of abstractions we engage with? The closest example here would be the [29:05] attempted solution of the abc conjecture. We maybe shouldn't get into that one, [29:10] but it probably is not a correct solution. Basically it's this whole new way of [29:18] thinking that this otherwise reputable  mathematician in Japan had come up with. [29:23] It took mathematicians a long time to even  parse what he was saying, but it had the feeling [29:27] of an alien bit of mathematics that's theory  building, not just a long chain of reasoning. [29:34] He called it inter-universal geometry. The biggest fear would be that an AI does [29:41] that, and then much like the abc conjecture,  people work for years to go up the mountain, [29:44] and they're like, "Dang it. This just isn't right." If it turns [29:48] out to be wrong, but it really looked right. Even if it was right, there's just a lot of [29:52] effort to hike up a new mountain. If we end up in that situation, [29:55] David Bessis had a really great blog post  called "The Fall of the Theorem Economy". [30:01] He's talking about how historically, as  you were saying, mathematics is coming up [30:08] with these definitions and problems, and  it's about proving theorems about them. [30:13] The theorem-proving stuff is what gets all  the credit, but it's really a parasite on [30:17] the coming-up-with-the-definition stuff. Historically, this has not been a problem [30:20] in terms of credit apportionment, because if you  come up with a definition, you're probably going [30:25] to be the guy who comes up with a theorem. But now we're in a situation where [30:30] if the valuable work is coming up with the  insight and AI automates the latter part… [30:38] Imagine a scenario where an AI comes up with  Abel-like direct arguments about a bunch of [30:43] important conjectures in the world,  and then we just have these proofs. [30:47] Now it's up to humans or  future AIs to consolidate. [30:55] Again, having no object-level understanding of  this argument whatsoever, I'm sure that if you [30:58] had access to it, it would make it easier  for you to think about what's going on. [31:03] Is there some deeper way in which we  can understand why this proof works [31:07] that would make it easier to come up  with the ideas behind group theory? [31:11] I think it would be hugely helpful. So much of trying to discover new [31:16] math is mostly being wrong. You're trying to solve a [31:20] problem, and it doesn't feel like constantly  taking the correct step up the mountain. [31:25] Mostly it feels like a random drunken  walk, where you're doing a thing and then [31:28] you're wrong and constantly discovering that. If at the very least you know that trying to [31:32] digest what you have is ultimately leading to  a correct solution, that feels like progress, [31:37] simply because of the sense of  knowing it leads to a solution. [31:43] There are plenty of instances in the recent  history of math where it feels like the reach [31:47] has exceeded the grasp, where things are  proven long before they're understood. [31:53] One of my favorite openings to a  paper—it's not even a research paper, [31:58] it's more like an expository one—is from a  mathematician named Timothy Chow, who was [32:03] trying to understand a concept called forcing. There's this problem called the continuum [32:08] hypothesis that more or less asks: you have  a size of infinity for the natural numbers, [32:13] and a size of infinity for the real numbers. Is there something in between? [32:16] The answer is both yes and no. It depends on your axioms. [32:19] It's outside the scope of our usual axiom  systems, which is an interesting answer. [32:23] But the method to describe it  is really hard to understand. [32:27] It's this thing called forcing. In the beginning of this paper, [32:30] he writes, everyone knows the idea  of an unsolved research problem. [32:34] I want to propose the idea of  an unsolved expository problem. [32:38] Sure, we've proven it, but we  don't really know why it's true. [32:41] Then he proposes a partial solution  to that expository problem. [32:45] You can imagine why I loved that  framing, because this is my whole life. [32:49] I don't do research math. It's wholly about what's the most clear [32:53] way to understand this, even if it's proven. There is a difference between proof and [32:57] explanation, and I think you're getting  at the importance of that distinction. [33:04] Yeah. That will be the main incentive. Or the incentive would have to change, not just [33:09] in mathematics but in other areas of science, from  proving things about the world to consolidating [33:16] proofs into problems or higher-level insights. We were having a discussion earlier at lunch about [33:24] a recent talk you were giving on design and  how it helps us understand things. [33:31] In the limit, is there really a difference  between the conceptualization of an [33:36] idea and the idea itself? If you think about special [33:39] relativity and spacetime diagrams, and Minkowski  spacetime, this is a way in which we illustrate [33:47] why there's length contraction and time dilation. But that is the reality… So, the exposition does [33:56] seem to be the explanation in some sense here. There's a couple of interesting things there. [34:00] One is that there seems to be a really strong  correlation between the people who come up with [34:05] genuinely novel insights and the people who are  actually quite clear in their communication of it. [34:10] You might imagine the opposite, given that  the experience of a university student is [34:14] often that the expert teaching them is not  necessarily the best explainer of that topic, [34:19] because they're so spoiled by their expertise. But what seems, at least in some cases, [34:23] to be the case is that the people who are really  coming up with something quite novel—you've got [34:28] Einstein or Claude Shannon or someone—you  read their papers, and they're really lucid. [34:33] It doesn't feel like this is just for the experts  and you have to chop through it with a machete. [34:38] They're very good expositors. Feynman has this  characteristic too, he’s a very good expositor. [34:43] Maybe the same part of the brain that  comes up with the correct new way [34:48] of thinking about it at a research level  also has this knack for good explanation. [34:52] I think this is pertinent to AI. I used to think that AIs would [34:58] become these automated theorem provers, but  the role of mathematicians was going to shift [35:02] towards my job, explaining these things. Now I suspect that actually they'll also be [35:08] quite good at doing that, probably better than  most humans are at explaining and distilling. [35:13] So digesting and explaining what was  going on is probably actually not [35:15] what's left for mathematicians, by the  nature of how these things are going. [35:24] We can talk about ways this might not be it,  but probably the same thing that comes up with [35:28] the really good new idea that solves some new  problem is also just good at explaining it. [35:35] That's a way my beliefs have changed. What's the last thing you think you'll be doing? [35:40] Both you and also what the human  mathematical community will be doing. [35:45] I will probably be doing something  like what I am until I die. [35:53] If the doomers are right, maybe  it'll be for the same reason. [35:58] Yeah. You build a man a fire,  and he's warm for one night. [36:02] But set a man on fire, and he's  warm for the rest of his life. [36:05] So that's where I am with AI. Some of the function of an [36:11] explainer or a teacher is to add clarity  to a thing that someone's curious about. [36:16] That's one thing. But some  of it is more relational, [36:21] providing motivation and a sense of curation. One interesting take that I've heard about [36:27] what mathematicians will end up being  is that it's actually more analogous [36:30] to art museum curators than anything else. The AI solved the thing, so the art exists. [36:36] They even know how to explain it really well. But you still want someone to help you navigate [36:42] this nearly infinite space of what  ideas are worth engaging with. [36:47] Even if AIs were in some sense better at  that, I think we would always still prefer [36:52] a human that we had a relationship with,  because the way we get motivated to be [36:56] interested in things is a social phenomenon. If you have some specific technology you're [37:01] trying to build, that might be different. But the people listening to this podcast [37:07] trust your curation on what's an  interesting topic in the first place. [37:11] It's not that they're landing here because  whatever your next topic is, that's what [37:14] they wanted to understand in a prior sense. They're trusting you as a curator. [37:19] So my role, and arguably that of other  mathematicians, might actually just shift [37:24] subtly into that curation direction  of what ideas are worth pursuing. [37:28] That's a lot of my job right now. I think people assume a lot of [37:32] the time for a video goes into the visuals. Sure, it does. It’s not immediate. But actually [37:38] a lot of it is just deciding what's worth saying  in the first place, what's worth putting there. [37:46] I want to engage with that, and I think  I have a trust with certain people, [37:49] and they're curious what I would choose to put  forward even if the AIs are better than that. [37:53] It's the same reason human musicians  are always going to have a role: [37:57] that social function of the story behind them,  even if the objective quality of the MP3 file [38:02] coming out of some model is better. That's what I see happening to my job. [38:08] I want to go back to a question from earlier. Just as AI has crossed this threshold, [38:15] this important benchmark of being able to  connect existing ideas to come up with a [38:19] new discovery or prove or disprove something,  we're like, "Okay, but what's the next thing?" [38:27] There's a lot more to do on that one, by the way. Just because a couple lightning bolts have been [38:31] thrown… I think there's this flourishing future  over the next couple years of really connecting. [38:35] Right. So in the limit, you could even  say—I don't know if this is accurate, [38:40] but potentially—a lot of the biggest  breakthroughs look like this at some level. [38:46] With general relativity,  you're just connecting together [38:52] Riemannian geometry and special relativity. So as AIs keep getting better and better at this [38:56] connection thing, maybe a lot of big breakthroughs  are not really of a different qualitative nature. [39:02] I don't know if you have a take on that. A lot of the conversation has focused on [39:07] problem-solving and that nature of math,  ticking off Erdős problems or something. [39:11] But I'd say it's not even a majority of  mathematicians who would characterize their work [39:15] as really targeting the next problem to tick down. Are you familiar with the Langlands program? [39:21] No. It's not even a [39:23] field of math so much as it is a research ethos. Fermat's Last Theorem is one inkling of this. [39:30] You had these two seemingly disparate things,  and a connection between them led to a solution. [39:37] Langlands was a mathematician. He has  this famous letter essentially spelling [39:41] out how it seems likely that there's  a lot more connections like that. [39:46] He even got a little bit more specific  about the nature of the connections, [39:49] such that you might imagine this  large map, and you've got this [39:52] valley over here and this mountain over  here and this set of plains over there. [39:57] There's a lot of mathematicians who would  characterize their work as being part of [40:00] trying to understand the threads on this map. The progress there, it's not even "Here's this [40:05] one specific problem that we know  will be solved by that connection." [40:08] It's more that time and time again, there have  been cases where big problems were knocked [40:13] down by finding connections, such that it's  almost preemptively finding the connections. [40:19] It's actually very interesting.  Anytime you run into a mathematician, [40:25] ask them whether the character of their  work is more akin to the Langlands program [40:30] or to targeting one particular problem. You get a certain bifurcated split there. [40:36] The possibility of AIs being supercharged  connectors feels like it might be an [40:43] amplifying tool in that pursuit. It's hard to measure, though. [40:49] This cuts to what we were saying earlier: How do  you assign a score to say, "Yes, you've done it"? [40:55] If it's knocking down a problem, you have a  clear way of saying, "Yes, you've done it." [40:59] You can write the headline. You can have your PR move as [41:01] the AI company to say, "We did it." Whereas if it feels like that was the [41:05] right connection to draw, you  can write theorems around it. [41:08] That's the nature of what the  papers in that field look like. [41:10] But I think it will require a lot more  "human in the loop" to say, "What was the [41:16] kind of connection that we were going for?" That's my guess on what most of the useful [41:22] progress from these models will  look like in the next five years. [41:26] It's just really filling in that  landscape of connections that you [41:29] can draw if you're an expert in multiple fields. As you've pointed out, it's kind of surprising [41:33] we haven't already had this. I would be curious to know at [41:38] a technical level what causes the unlock there. On the one hand, you can paint an explanation in [41:46] your head for why you could be an expert in all of  these things and not be drawing those connections. [41:51] When the method of reasoning is this  autoregressive chain-of-thought phenomenon… [41:58] Autoregression is actually a really weird  way to produce stuff, if you think about it. [42:06] You're an intelligent person. Imagine I've  locked you in a box, and the only way you [42:11] have of interacting with the world is that  you receive a slip of paper, and someone says, [42:15] "Can you predict what will come next?" You predict what will come next, [42:18] and then your memory's wiped. You get another slip of paper. [42:23] Imagine that was done a whole bunch of times,  and then what comes out on the other end. [42:26] They say, "Look at this essay that you wrote." You might look at that and say, "This is awful. [42:30] That's not the essay that I would've written." The process of repeatedly predicting something [42:34] is just pretty different from how you would think  as a writer to compose it and think it through. [42:40] In particular, what would probably happen  is that you're a slave to your context. [42:45] You might be answering some question  about a particular field, so you draw [42:49] on all the context around that. But the connection where all the [42:54] substance is going to come from is,  by its nature, a very unlikely one. [42:59] You can do all the RL that you want to try to  get better in some way, but what's the thing [43:03] that's specifically upweighting and incentivizing  making these unlikely connections when the vast [43:07] majority of them aren't the predictable  next token that would come in there? [43:12] So it might be the case that you just have  this intelligence locked inside that box, [43:17] but it's a weird way of interacting with it. The thing I'm curious about is: [43:21] do you ever get any fruit by questioning  the premise of how tokens are generated? [43:30] I don't think it would be as simple  as manipulating the temperature, [43:34] but are there any things that you can do that take  the existing level of intelligence but find the [43:40] right ways of sparking those connections that  unlock these sorts of things that we've seen? [43:46] Or do you just need a little bit more  intelligence, such that at the level of [43:50] prediction, it's predicting that it should be  making that lightning bolt to another field? [43:55] I think it's more productive to reason, instead  of architecture or even loss function, about data. [44:06] We have diffusion models that do text,  and the kinds of things they produce [44:11] are not of a wholly different character. They've just not been explored as much. [44:14] I think the more relevant thing is: what is  the data on which whatever architecture or loss [44:20] function you have is incentivizing you to produce? It does seem like they're getting better. [44:27] Forget about math. We did have a couple of  examples of this kind of thing, but if you [44:32] just look at why they're getting better at being  autonomous agents… They're in an environment where [44:39] they’re autoregressively producing the step that  says "Let's step back and do a search over the [44:44] whole codebase," and then "Let's step back and  assess my mistake," is the thing that works. [44:50] I assume what happened in the case  of progress in science or maybe in [44:55] math is you have frontier math-like problems. Mathematicians have specifically designed them [45:02] because they require connecting  together two different fields. [45:07] I'm guessing there's all kinds of clever,  partially synthetic ways to make harder and [45:10] harder problems like that that require  these kinds of connections—for example, [45:14] by eliminating assumptions and still requiring the  AI to get to the answer—and then it doesn't really [45:21] end up mattering what the loss function is. It's really about, can you come up with an [45:26] environment that incentivizes this ability? It feels like you should be able to. [45:32] I certainly can't speak to the correct  ways of doing that to unlock all this, [45:36] but it would just be pretty surprising. Don't you think it would be surprising if, [45:40] over the next three years, there weren't  a lot more of those lightning bolts? [45:43] I think this is an important thing to think about. We often think about how smart a single system is. [45:48] And we don't think about AIs having  advantages that are more the result [45:54] of other facts about them. So in this context, [45:59] the key fact about them is that we can just  parallelize and arbitrarily scale them. [46:04] Whatever level of capability they have,  it's not just one idiosyncratic genius [46:09] in the history of mathematics who makes a  few connections and then dies in a duel. [46:13] It's universally applying that waterline  across all problems that are accessible [46:18] at that level of capability. This is among the many advantages [46:22] that digital minds inherently have  that we don't think enough about. [46:26] The other ones being that they can merge all  their knowledge together—or at least that [46:32] there will be techniques that allow this to  happen—and that you can spawn off copies with [46:36] identical levels of knowledge. This parallelization is [46:40] quite an important property. I'd be curious about your predictions. [46:43] Even if they're not as smart as human  mathematicians, the fact that for PR reasons [46:49] the AI companies are just throwing billions  and billions of dollars at this means that [46:55] quantity has a quality all of its own. That seems in the right direction. [47:00] If we take that conversation between Montgomery  and Dyson at the IAS that suggests some connection [47:07] between the Riemann hypothesis—or the Riemann  zeta-function zeros—and random matrices, [47:12] that feels like the kind of thing  that you could try to automate. [47:16] You have agents representing  expertise in all these fields. [47:21] We all know that an institute  is smarter than an individual. [47:24] The reason for having people all in the same  geographic location is that you want those [47:30] serendipitous conversations to happen. What does it look like to engineer [47:34] those between agents? It's interesting, because you [47:37] point out that you can pool all your knowledge,  but I really wonder if one of the advantages is [47:41] that you can do the opposite of that. Sometimes when an AI is failing, [47:47] it's because it gets into a bad chain of  thought and it's really hard to get it out. [47:51] So you say, "I'll just start again." Same deal with humans. Sometimes you [47:54] start thinking about it in a certain way,  and what's required is to just back up. [48:02] There are stories about people trying  to prove something for a long time, and [48:04] then at some point they say, "Hang on a second. What if I tried to prove that it's impossible, [48:08] or prove the opposite?" Unwinding your own context and [48:12] going at it with a fresh mind… You could imagine  systematizing that, or having multiple different [48:16] agents deliberately given different pieces of  context and trying to compare and contrast there. [48:21] We don't have the same level of  manipulation on our own context. [48:26] In this AI and math series, the first episode  we'll do will be about when they solved the IMO. [48:32] I want to focus on one specific IMO problem  that they failed on, which is one that a lot [48:36] of very smart students failed on. Terry Tao also failed on it. [48:43] People were very mad at the problem  because they called it a troll problem. [48:47] I almost don't want to spoil it, because I  want to construct the episode around leading [48:51] someone in without their knowing that  it turns out to have a simple solution. [48:56] You can really empathize with what  it's like to be a student solving this. [49:01] Basically, there's a really elegant way of  going down what you really feel like is going [49:05] to be the solution based on the context of it  being an International Math Olympiad problem. [49:11] The character of the solution is really enticing,  but it's hard to prove that it's the best. [49:15] The reason is that it's not. There's this almost brain-dead [49:17] solution that is the best. The relevance of that to the whole [49:21] AI story is that for a human, what's required to  answer that question is to escape your context. [49:27] Escape the context of being in the IMO. Escape the context of the way you've been [49:30] trained to solve these contest math problems. If you just approached it like a brain [49:36] teaser that I throw at someone off the  street, they'd probably answer it well. [49:40] You want the same sometimes for human research in  other contexts, just being able to refresh your [49:50] thinking and come at it completely differently. Of all the advantages that digital minds have, [49:55] that might actually be one of them: a more  systematic approach to refreshing your thinking. [50:02] Spin off two agents, one who's trying to prove it  and one who's trying to disprove it, one who tries [50:05] it this way and one who tries it another. They deliberately have different contexts. [50:10] I would be curious to see, if we're having  this conversation three years from now, [50:13] how many of the significant results that make  headlines have that character of basically [50:19] erasing the context previously, trying a bunch  of different things as opposed to merging [50:23] the results of a bunch of different agents. It is incredibly interesting, because a common [50:27] concern people have about AIs is this entropy  collapse where they all think the same way, [50:32] because they're trained in similar ways. This is why they're bad at writing. [50:35] They go down the same path and have  similar patterns of speaking and so forth. [50:40] But maybe the key advantage AIs have is that you  can systematically… It sounded like one of the [50:48] reasons the unit distance problem conjecture took  so long to be disproven was that people assumed [50:53] the conjecture was actually true, so they were  mostly trying to figure out ways to prove it. [50:58] Maybe one of the key advantages the AIs will  have is to increase the entropy by systematically [51:06] trying out both the negation and trying to  prove the positive of any given statement, [51:11] or being able to systematically give  different agents different biases. [51:17] It seems like an important thing in the history  of human science is that Einstein was really [51:20] motivated by this bias that things should  look the same in different reference frames. [51:26] He had multiple other biases like these, but  that one was very formative in his thinking. [51:30] You can systematically survey a  bunch of heuristics and see which [51:34] ones are being productive on a given problem. [51:36] So you would suggest systematically  increasing entropy at the prompt level [51:43] even though you have this inevitable  collapse at the autoregression level? [51:49] Einstein would be an interesting example, because  he's got this bias toward things being relative. [51:53] He also has a bias toward  "God should not play dice." [51:57] You want to make sure you don't accidentally  have all your LLMs be Einstein, because you [52:01] might halt progress on quantum mechanics. Which goes to show you that there's not [52:05] a correct heuristic for science. You just need multiple independent [52:08] research programs with their own heuristics. That feels like old-school software. [52:13] As long as you're able to  describe that in some way. [52:15] You have old-school software  that amplifies that entropy. [52:19] If you're able to put a clear ontology to the  distinct ways of thinking that you want to prompt, [52:26] you explore that full ontology, and then each  individual one runs off doing what it is. [52:33] There's a certain design question there about how  exactly you describe the different approaches. [52:38] The easy one is: are you trying  to prove it or disprove it? [52:40] The harder one would be to say, what are  all the tactics you could take to prove [52:44] this, and make sure you're applying  sufficient breadth to exploring them. [52:50] I don't think people appreciate the  kinds of things these models can [52:53] just go handle for you when you equip  them with a good harness like Cursor. [52:57] For example, I started publishing my episodes  on Bilibili for a hopefully burgeoning Chinese [53:02] audience — but everything I upload there  needs the sponsored segments cut out. [53:06] Normally that would have meant asking my  editors to go back through all the old [53:10] episodes, cut the ads, and re-export everything. But in about as much time as it would've taken [53:15] me to send them that Slack message, I can just  tell Cursor to do it instead and spare them.And [53:19] for podcast research, I have a whole repo where  I've put every single book and paper that's been [53:25] relevant to prepping any of the recent episodes. I've been able to just throw everything in there, [53:29] because the Cursor harness is extremely good  at helping the model figure out exactly what to [53:34] pull — whether from my repo or from the web — to  answer the questions I have while I'm researching. [53:40] So whatever you happen to be working on  right now, just try pointing Cursor at it. [53:44] Go to cursor.com/dwarkesh to get started. Obviously, AI in math is making much faster [53:50] progress than everything else, and  people point to the verifiability of [53:53] the domain as the key reason this is happening. I think that's one of the two important reasons, [53:59] but people really neglect the other one. I'm outside the labs, so I don't [54:05] know what's actually going on. This is a totally naive theory. [54:12] A tangential question to why AI is  making so much progress in math: [54:16] why has it been so slow at computer use? A computer is very verifiable. [54:21] Is my Etsy package coming? Is my event booked? These [54:25] are extremely verifiable things to survey. What computer use lacks is grindability. [54:32] Because websites have bot detectors—and it takes  a tremendous amount of compute to run parallel [54:37] rollouts—it's very hard to run a thousand parallel  rollouts of the same checkout flow on Amazon. [54:44] You'll get shut down by Andy Jassy. Him personally. He presses the red [54:50] X on Dwarkesh button. Exactly. You could try [54:54] to build clones of every single website, but  that's very labor-intensive and slows you down. [54:59] The reason you currently need to do so many  parallel rollouts to learn a skill with [55:04] deep learning is that we haven't  solved sample efficiency. [55:08] Sucking supervision through  a straw, as Karpathy says? [55:10] Exactly. Of course people are working on many  different techniques, but fundamentally there's [55:13] this big constraint in the way we train AIs. With code, you can containerize a given [55:21] level of progress in a repository and then spin  out hundreds of parallel containers and say, [55:28] "Try to implement this feature,"  and it's totally deterministic. [55:31] Because it's deterministic, you can solve the  credit assignment problem because you know that [55:35] whatever caused this rollout to succeed and this  one to fail, the diff is the thing that worked. [55:41] If you have situations that are starting off at  different starting points, this credit assignment [55:46] problem becomes much harder to solve. Most things in the real world are [55:50] very hard to containerize in the same way. Coding and math are exceptions to this rule. [55:55] But if you're trying to figure out how to  build a new business that succeeds, or how [55:58] to go trade in the markets for a day and make  money, the fact that you have to interact with [56:03] the real world and things change day after  day means that you can't keep replaying and [56:08] grinding and farming the simulator. Math, of course, is the exception, [56:12] and I feel like this is an important driver  of progress in this domain and also in coding. [56:20] It's not just verifiability;  it has to be grindable. [56:23] The third reason people point out that AI is  making fast progress is they focus a lot on [56:29] Lean and formalization. Again, I have literally [56:31] no idea what's going on in the labs. I feel like Lean just doesn't matter that [56:35] much for the current level of progress in AI. Why is AI able to disprove the conjecture [56:42] about the unit distance problem? They released the chain of thought, or at least [56:46] a rewrite of the chain of thought. It didn't have any Lean in it. [56:50] I think the process-based supervision that Lean  provides, where you know each step is correct, [56:55] seems less relevant than just having this  grindable outcome that is verifiable. [56:59] It's an interesting point about  grindability mattering more. [57:06] Naively you might think Lean  provides something unique for math [57:09] because you're able to see if it can prove it. You have old-school software that can tell you [57:14] yes or no, and you use that as your VR. What would corroborate your [57:19] point is the initial attempts. Again, I'll circle back to the IMO. [57:22] Initially, DeepMind basically does that. Everything is in Lean, and then the next [57:26] year it's all in natural language. So to your point, it's not needed. [57:29] I do think there's a yet-to-be-explored  benefit of that formalization domain, [57:36] which is that at the moment you still need a human  reviewing that counterexample to the unit distance [57:43] conjecture to say, "Looks good." That provides a certain bound [57:48] on how endlessly explorable things are. If you consider AlphaGo or AlphaZero-style [57:53] systems, they're off in their own universe  playing a bunch of Go and exploring themselves, [58:00] potentially going off the rails of what  any human needs to look at, but they still [58:03] have this automated verifiable reward. It's not just that you can do RL on that. [58:09] It's also that you basically never  have to check in, and you can just pour [58:13] compute at them exploring the universe of Go. What stands to be interesting—maybe this won't [58:18] pan out, but the jury should still be out on  whether it'll yield anything—is that with Lean, [58:25] you could imagine having a basically  endlessly running program that's [58:31] constantly trying to extend Mathlib. Mathlib is this GitHub repository that's [58:35] basically all of math written in code. It's very far from all of math, but they [58:39] want it to be all of math. It’s written in code where [58:41] you can ask, "Is this proof correct?" It's very labor-intensive to write these proofs. [58:45] There's a whole subcommunity around it. But you could imagine having an AI where [58:51] you say, "Simply try to extend Mathlib." Maybe it's a fork of it so that it doesn't have [58:56] trash in it, because people have a certain  taste for what they want to be in there. [59:01] So you have your fork of the pure AI Mathlib,  and it just goes and it doesn't stop. [59:05] It doesn't need anybody to check in on it. It could just keep going. [59:09] It might come up with its own conjectures. It might come up with its own [59:12] theories and different definitions. Maybe many of them are useless, but it [59:15] just has this infinite tree that it can grow out. That's a very unique thing that math has that [59:20] nothing else has, where you could press go and  just pour compute at it, look away for ten years, [59:26] and then come back and say, "What do you have?" There's going to be something. [59:30] Then there's a question: is it useful or not? How do you suss that out? [59:33] That's just an interesting thing to be able to do. It would be very surprising if that didn't [59:37] yield some sort of interesting  mathematical insight from it. [59:44] There are two different ways that  Lean is important in this story. [59:48] The first one is how you could let go, not  even check in, and progress will be made. [59:54] You can do that with Go. I don't think you can do [59:56] that with natural language math. That's very interesting. Did you [60:00] see Karpathy's auto research idea? He wrote this one Python file that [60:07] does basic LLM training, and then had a  repo where LLM agents would try to make [60:13] modifications to the file, and if it sped  up the speed run, the modification stays. [60:18] Eric Jang, who came on to explain how  AlphaGo works, did a similar thing when [60:22] he was trying to build a very strong Go bot. He had interesting observations. It's really [60:29] good at running an experiment and going  down that path, but it's bad at stopping [60:34] at dead ends and doing extremely parallel things. Anyway, this will probably change in the future. [60:41] It's very interesting to think about  what it looks like in the limit. [60:44] This is fundamentally what the human  institution of mathematical research is. [60:50] It's a library extended in  interesting and useful ways. [60:54] This way you don't have any  outcome-based supervision. [60:57] There's no outcome that you're trying  to incentivize, but you have a process. [61:00] You know the steps are correct, you just don't  know if it's going in an interesting direction. [61:04] If you were doing that, you don't want  to completely go off the rails and do [61:07] a random walk through the space of logic. You'd probably want some supervisor model [61:11] that's trying to provide heuristics  on whether it's useful or not. [61:17] You know people are working on it. That's one of those "five years from [61:21] now" things where I'd be curious to get  the future version of us talking about it. [61:26] Maybe that goes nowhere, but Terry Tao  was talking about one research project [61:32] that tries to exhaustively search  the space of possible algebras. [61:36] You could imagine different axioms  that you apply to algebraic systems. [61:40] When we come up with group theory, there's a  certain axiom system that looks like arbitrary [61:45] rules unless you know the motivation. What if you tried all of them? [61:49] Do any of them yield useful things? The vast majority of them [61:52] are just trash in some way. It all collapses to no interesting results. [61:55] But every now and then, there would be this little  island of a completely different type of axiom [62:00] system that at the very least seems rich in terms  of the number of theorems that can come out of it. [62:05] That's bread and butter for what you would  imagine automated provers being good for, [62:09] exploring that space and seeing which  one of them turns out to be something. [62:13] Maybe one of those islands actually turns  out to be something you can retroactively [62:16] put motivation on, to say this is the  kind of structure it's trying to get at. [62:19] In the same way that you could imagine looking  at the axioms for a group, not knowing that it's [62:23] about symmetry, but you retroactively realize  this is very relevant to studying symmetry. [62:29] You could imagine results of that flavor,  but instead of just exploring possible [62:33] algebra systems, it's exploring all possible  logical consequences of any kind of axiom. [62:39] On the point about whether you can provide  process-based supervision without Lean, [62:44] DeepSeek had their DeepSeek Math model. They released a paper on how they trained it, [62:51] and it was quite interesting. The problem with natural language [62:54] proofs is you don't know if it's correct or not. They have a verifier, and the verifier is trained [62:59] by a meta-verifier that makes sure that for  all the problems they're training this model [63:04] to solve in the art of problem-solving,  the verifier is giving good feedback. [63:09] It works. It's interesting that natural language  verification with some sort of meta-verification [63:15] seems to work so far in the published literature. It also seems to work in the published [63:19] products that we're using. If you look at coding agents, [63:23] they're getting better and better at  writing clean code and refactoring code. [63:28] I'm sure there are process-based "LLM-as-a-judge"  systems providing taste and saying, "Is this a [63:35] clean way to write this function? Are there duplicates of [63:39] the same kind of modular forms?" That should also work for mathematics, right? [63:46] It seems more plausible for math than anything  else, even if you're only working in natural [63:52] language, that you could trust a verifier. You and I were talking earlier about [63:55] why they're bad at writing. They seem to be good judges. [64:00] If I give them two essays that  students wrote, they'd be able to [64:03] say which one is more accurate and insightful. So why can't you just have a verifier saying, [64:08] "Is this a good piece of writing or not?" Maybe the ultimate failure there is that [64:12] even if they're good at discriminating between a  B essay and an A essay, they're not actually good [64:17] at discriminating between an A essay and a thing  you actually want to read, something that would be [64:22] followable on Substack and insightful. They actually end up preferring [64:28] uninsightful pieces of writing. On the math front, the step to simply [64:35] know if a proof is correct or not lends itself to  an automated verifier, even in natural language. [64:42] You could probably still make a ton of progress. I still like the tree of logic out of Lean, [64:48] just in that you can really go off the rails. There's no constraint on the previous [64:52] way things had been phrased before. Everyone talks about move 37 in AlphaGo. [65:00] What is the thing that lends itself  to going outside the prior heuristics? [65:05] It seems productive to have a disconnection  from the rest of the world in that exploration, [65:12] as a complementary research pursuit  to the natural language math front. [65:17] The other relevance of Lean  would be, let's say you have your [65:21] pure natural language RL environments and  a pure natural language set of proofs. [65:28] People say, "Proceed, AI mathematicians,"  and they generate ten papers a day. [65:38] If there's any error rate to that at all…  Alex Kontorovich has talked about this. [65:43] It becomes insufferable as a mathematician. Every single time you see one of these, [65:49] you don't know if it's worth your time. Even if 99 out of 100 are right, I don’t [65:54] know if it’s worth my time because it's really  labor-intensive to find what that error would be. [65:59] It's really frustrating to spend all  your time on a paper that was trash. [66:04] Having something that's able to give  you that green checkmark that says, [66:07] "Even if this is going to be complicated to  understand, even if it’s going to be a pain, [66:11] you at the very least know it is correct,"  every other field would kill for that. [66:16] Math has that. If the models are also able to take  their natural language proofs and formalize them, [66:22] that seems huge. Every field would [66:26] love to have something like that. So I think you're right that Lean is [66:32] maybe overrated regarding its importance as a  VR environment for progress in math generally. [66:40] But I definitely wouldn't  write it out of the story. [66:44] I also love this extension of Mathlib  as a metaphor for what's going to [66:50] happen to our civilization pretty soon. For millennia, humanity has built this corpus [66:57] of knowledge and understanding, and everything  that we have is now distilled into these models. [67:02] At some point, the models will  just extend that arbitrarily. [67:07] By the way, on the writing front, I have  a theory of why writing is making worse [67:13] progress than these other domains. One reason is what you said, that [67:16] they're bad at judging not only A versus  B, but they get totally derailed by B*, [67:23] which is this shitty essay that hits all the  bells and whistles that A is supposed to hit. [67:29] The reward hacking thing just goes off the rails. But the other important thing is that writing [67:34] is not modular in the same  way that code and math are. [67:40] You can write a function many different  ways, and they do the same thing. [67:43] Of course you want it to be clean, but at  the end of the day, if it works, it works. [67:46] Same with lemmas in mathematics. [67:49] You can have some end product that's  different from the way it's produced. [67:54] Code is the thing that produces some end  product, and you want a functional end product. [68:00] Whereas in writing, the end product is  directly the thing the AI is producing. [68:05] Each paragraph, sentence, and word  matters because that is the substance. [68:13] It's not some separate thing  produced out of the writing. [68:18] It can't be slop in the way that code can be  slop and still produce the outcome you want. [68:25] But you were just pointing out how we've  actually gotten much better at agents [68:29] writing not just functional code, but clean code. Why is it not the case that the same progress [68:34] that lets you go from merely functional to a clean  and mergeable PR also results in clearer writing? [68:42] That's a good point. Also, has it  not? I agree there are many ways [68:46] in which they're terrible writers. But for a lot of writing I consume, [68:50] I find it's better to just copy-paste it  into an LLM and say, "Explain this to me." [68:56] The explanation will be better than  the thing produced by the human. [69:00] It's funny that we say these are such  terrible writers, and yet my revealed [69:04] preference is to have an LLM explain it. Even when I'm talking to a human expert live [69:09] on a call, if it's a piece of knowledge that only  they have that's not encoded in the distribution, [69:15] I want them to explain it to me. But if in order to understand that, [69:18] I need to understand a more basic concept, I  would prefer if it were socially acceptable [69:22] for me to just say, "Let's pause here. I'm just going to ask an LLM how that works, [69:26] and then we can come back to  your special piece of knowledge." [69:32] That's distillation, an explanation.  If I'm thinking of your quality as an [69:39] essay writer—if I give you a book to read  and I want a book report—I might believe [69:44] that the LLM gives me a better book report. But what people are really getting at when they [69:51] say it’s bad is, what is writing? It's not just the distillation [69:54] of preexisting ideas. It's not just how you explain clearly, [69:57] because they are good explainers. It's about what the insight is. [70:01] This is where autoregression is a  very weird way to generate things. [70:06] When you're writing, you sort of know that in  order for it to be good, you have to have an [70:11] element of the unpredictable. It's not just [70:14] increasing the temperature in your mind. It's knowing exactly the correct point [70:18] when you want to make an unpredictable move, and  that that's going to be what's more insightful. [70:23] Even if it's better at explaining a preexisting  thing, what generated that book that you wanted [70:27] distilled in the first place? It wasn't an LLM that generated [70:31] it and you just needed it. It was some author who, [70:34] through a lot of exploration of ideas in the  world, decided what aspects were interesting [70:38] and what ways of presenting it formed  a coherent, well-motivated narrative. [70:45] They put that all together in some way. If they're a good author, you would [70:50] probably err on the side of reading  their book instead of the distillation. [70:53] Still, what makes it worthwhile to explore at all  in the first place and want to upload it at all? [70:59] It's that side of it that people cite  when they say LLMs are bad at writing. [71:04] It's that element of unpredictability,  of deliberately choosing something novel [71:08] that is very directly contradictory to  the way things are typically produced. [71:13] That's a good point. I think they're  also really bad at building really [71:16] good mental models of people, which  is a very important skill in writing. [71:21] Andy Matuschak and another collaborator,  whose name I'm forgetting right now, did [71:24] an interesting report where they tried to teach  LLMs to write good spaced-repetition prompts. [71:29] I really like this because even though  it seems like a totally random skill… [71:34] It's just like, people are talking about recursive  self-improvement in a year, and we can't get [71:38] these things to write good flashcards. What's going on there? They tried many [71:45] different kinds of techniques,  and they're sophisticated people. [71:48] They tried to RL open source models. They tried all kinds of things, including [71:52] chain of thought and a big prompt they  sent to the best closed source model. [71:56] The key constraint, it seemed to me, was  that writing a good card is about projecting [72:03] somebody's mind in three months. What is the way in which they'll [72:08] associate the question? What kind of answer will [72:10] they be thinking at that moment? Is the elicitation that inspires [72:15] the detail you actually want to take away from  the passage you're trying to make cards about? [72:21] I think writing is similar to this. If you're writing something, [72:25] the reason it's such an enervating process  that takes so long is that with each word or [72:31] each sentence, you have to be thinking: what  is happening in my reader's mind right now? [72:35] Even if I flip the phrasing around so  the end phrase goes to the beginning [72:38] and this is the first image that comes to  your mind before you read the rest of the [72:41] sentence… Maybe autoregression is bad at that. This is maybe a more diffusion-like property [72:48] of considering the whole rather  than going sentence by sentence. [72:51] But also I think that  requires a lot of mentalizing, [72:53] which these models weirdly struggle at. It's an interesting question. Is it [72:57] weird that they struggle at that? I might butcher this. You know how [73:02] you cite studies that you once read  and maybe the study wasn't real? [73:06] There's one very memorable one. Let's say you want to quiz people's EQ. [73:14] You show a flashcard of someone's  facial expression and someone [73:18] is trying to describe that emotion. There are really good tests online that have [73:23] a face and then four possible emotions. It's surprisingly hard to describe exactly [73:29] the correct emotion, but you also get the  sense there really is a correct answer. [73:32] If you try this with people in your life, you'll  notice that the ones who are pretty plugged in [73:36] socially do really well on it, and the ones  who are a little bit more left-brain don't. [73:42] That is a kind of test you can do. I vaguely remember an experiment to this effect [73:46] where they took people who had freshly gotten  Botox, and they did a pretest and a post-test. [73:53] Post-test, they were just much worse  at reading people's expressions. [73:56] That feels weird. Wait, they got Botox? [73:59] The person taking the test. You do the test, and then you go and get Botox [74:03] and your face is all frozen, and now you're worse  at understanding the emotions of what you see. [74:08] The thought is that part of understanding the  emotion you're looking at is doing it yourself. [74:15] At a facial level, you're  moving your face muscles. [74:19] You see that, you mimic that, and you're  like, "Oh yeah, that's anxiety," at some [74:24] very subconscious level. So in that sense, if it is [74:27] the case that models have bad theory  of mind, sure, they know everything [74:31] because they've read what everyone wrote. But at the level of actually being able to [74:36] put themselves in your shoes in the same way  that my face muscles are mimicking your face [74:39] muscles—that's what helps me understand  how you feel—it's not surprising at all. [74:43] They don't have face muscles. Their brain works completely differently. [74:45] It's like an alien trying to empathize. How could it have theory of mind? [74:50] It would be this very emergent thing to have. Whereas we can just plug it into our own minds. [74:55] We've got the ready-made  hardware to just place it in. [75:00] From that lens, it's not that surprising. Okay, Grant, we're both partners with Jane Street. [75:06] I'm sure over the years you've interacted with  a lot of Jane Streeters — what have you found [75:09] that's unique about them, or their culture? I did this interview with them this year, [75:13] that was partially interesting because they  don't usually have anything outward-facing. [75:17] I mean in the industry they're known for having  a pretty wild retention rate — people just stay [75:21] there — getting an inside view of that. I remember in one of their comments someone [75:25] saying: even though the people have  role titles — researcher, or trader, [75:29] or engineer — they often don't know what  their colleagues' actual role is, because [75:33] everyone's doing a little bit of everything else. Like even if you're officially a trader, you're [75:36] doing a lot of research; even if you're officially  a researcher, you're doing a lot of coding. [75:40] And I suspect that's part of why they  have the insane retention they do. [75:44] Because anyone who wants to grow just has the  chance to do a lot of different kinds of things. [75:49] All right, Grant, I'll do  the plug for you this time. [75:51] If you want to watch the full sit-down interview  Grant did with some of the folks there, [75:54] go to 3b1b.co/janestreet. All right, [75:59] Grant — let's talk more about AI and math. What advice do you have about using LLMs to learn? [76:08] As I was describing, for a lot of well-known  concepts, I find them very helpful. [76:13] But often, just a couple of messages further down,  I'm trying to understand something, and they're so [76:21] confused themselves that they're confusing me. They don't explain it the right way. [76:26] I know that talking to the right human could  clear up my confusion in three minutes. [76:32] More and more, we're going to  want to use these things to learn. [76:39] People talk a lot about education  and representation stuff. [76:42] Have you noticed ways to use them more  productively to understand concepts? [76:45] I'm curious to hear your take on this. I'll give mine. Even pre-LLM, [76:50] I feel like a relevant insight in learning  was recognizing who matters more than what. [76:56] My advice to any college student when they're  choosing what courses to take: care a little [77:01] bit less about your preexisting interests,  because they're kind of arbitrary right now, [77:05] and care a little bit more about whether  the person teaching it is a good educator [77:08] and someone you resonate with. In choosing what books to read, [77:13] who the author is maybe matters more  than whether it's a prior interest. [77:16] If there's a book you've liked before, read what  else that author has written rather than reading [77:19] another thing on that subject. I'm getting to LLMs on this. [77:26] There's a difference in feel for trying to  learn something from a Wikipedia page versus, [77:30] if it's a philosophy topic, going to  the Stanford Encyclopedia of Philosophy. [77:35] Or if it's a math topic, you go to the  Princeton Companion to Mathematics. [77:40] The difference there is the articles are  deliberately written by one individual who [77:46] tries to actually craft a motivation around it. Whereas on Wikipedia, it's this [77:51] local minimum that's reached where  every sentence has to be correct. [77:56] In a good exposition, you care a little  bit less about correctness on the way. [78:01] You can deliberately craft things that are a  little bit wrong that you correct along the way, [78:05] which gets edited out in a  crowdsourced environment. [78:09] LLM explanations feel to me at the moment a  lot like Wikipedia, which is to say, amazing. [78:15] Imagine a world before Wikipedia, how long  it would take to find and suss everything. [78:20] But nevertheless, what's the most  useful part of a Wikipedia page? [78:24] It's often just the references at the bottom. You look at the key references, [78:27] and you go to them, and you read them. Sometimes that gives a much better overview. [78:32] So often I like to just ask  an LLM, "Who should I read?" [78:38] Maybe I can even give some  specifics on ways I want to learn. [78:41] I actually got gaslit by this once when I was  trying to learn about semiconductors or something. [78:46] I felt it was a very visual topic,  but all the resources were text. [78:48] I asked, "Is there a well-visualized video  explaining the concepts you're getting at?" [78:57] And Claude said, "Yeah, here's a couple," and the  top one was like, "Here’s one from 3Blue1Brown". [79:02] I’m like, "I can guarantee that there’s not." It was an actual video, an actual link, [79:08] but it had just misattributed someone else's. It was good. I had a much better experience [79:14] clicking over and watching it to learn rather than  trying to proceed forward with questions there. [79:20] In that sense, I'm basically using it  like a very souped-up version of Google [79:23] to zero in on the right human-written resource. What about you? You engage with these a lot. [79:29] What's the best way to use them? I think you put your finger on it. [79:31] The most productive learning sessions  I've had are when there's some artifact [79:36] that a human has produced—whether it's an  article, a book, or a video—that organizes [79:41] the relevant concepts in the correct way. It builds up the motivation for why the next [79:46] idea would be relevant to solving  the next problem you'd encounter, [79:49] and the next idea, and the next idea. Then you use the LLMs to just do a [79:55] little bit of pruning around this  branch that the book has identified. [80:02] I was actually going through—I think you might  have recommended it—Steven Strogatz's textbook on… [80:08] The chaos one? Nonlinear Dynamics  and Chaos? I love that book. [80:11] Yeah, I was going through it, and it was bliss. It was like your videos in book form. [80:19] It was super fun. The way I was learning it,  I'd have his university lecture on one-third [80:23] of the screen, that part of the textbook on  another third, and an LLM on the last third. [80:30] I was actually thinking, if I were back  in college and watching this lecture live, [80:34] it would totally go over my head. These kids must be really smart, [80:38] because I'm pausing, reading the textbook,  talking to LLMs, and then restarting again. [80:43] But with him curating the right order  to understand concepts and the right [80:48] problems to motivate understanding them… Another thing LLMs are really bad at. [80:55] Something a really good human can do,  when you ask a question, a human can say, [80:58] "Actually, you're not really thinking  about this topic the correct way. [81:02] The question you want to be asking, the  correct way to organize these concepts, is X." [81:06] An LLM just can't really do that. It's a little too placating. [81:11] This is ultimately that sycophantic behavior where  it's very, "Oh, what an insightful question." [81:16] You want to strip that down. That's a good point, and I think [81:22] it cuts to theory of mind a little bit,  recognizing that asking a certain kind [81:27] of question reveals that the student's mental  structures are not the same as the explainer's. [81:34] Sometimes people do this to a fault. With a really good teacher, let's say [81:38] you have a middle school math classroom. If a student asks a question that suggests [81:43] they're thinking about it in a different way,  it's actually really hard to take that seriously [81:48] in the moment and ask, "Hang on, could you get  to a right answer with that?" before you say, [81:52] "Instead of that, let's do this." The really good teachers are able [81:56] to jujitsu the creative way the student  was thinking about it and bring it in. [82:05] LLMs aren't doing that. They  aren't reframing your question. [82:09] Instead, they kind of run off. At the very least, it feels [82:13] like there are three levels here. An LLM is at one, a good explainer [82:16] is at another, but the A+ explainer  is the one who can jujitsu your way of [82:21] thinking and say, "That's where that's useful." Maybe there is a cycle all the way around where, [82:27] five years from now, the LLMs will  be doing that, but in a better way. [82:31] What is your recommendation to students who  I'm sure email you this question all the time: [82:39] "I was curious about doing mathematics. I'm really passionate about the subject, [82:43] but seeing all the progress AIs are  making, I don't know if it makes [82:45] sense for me to pursue this as a career." This is relevant not only to people in [82:50] mathematics, but to anyone noticing that their  field is getting productivity gains from AI. [82:59] Coding is very adjacent to this. What advice do you have for people? [83:06] I wouldn't trust any advice that I give. That's how I'd couch it. [83:11] But even pre-AI, it feels very important for  any job you're going to go into to really [83:17] understand… If we're talking about a job—not being  a gentleman-scientist engaging with the math world [83:23] or something—you should understand where the money  is coming from, what value you're actually adding, [83:28] and the connection between those two. A surprisingly small amount of thought [83:33] is put towards that, especially by students. They're in this environment where they probably [83:37] want to go into math because  they've always been good at it. [83:40] They've been rewarded in life for  proceeding through the next hoop correctly. [83:44] When they think they want to be a  mathematician, it's because they think it's [83:46] a way to continue engaging with that. They think, "Where do people get to [83:51] do this?" rather than thinking, "What value am I  adding to other people, and to what extent is that [83:56] the reason a salary is flowing in my direction?" It's actually quite different in different cases. [84:02] In some cases, it's a very prestigious  mathematician, and their presence at a [84:06] university lends a certain brand value,  which is why the university wants them. [84:10] In some cases, an NSF grant is  given because of the public good [84:13] belief we have around basic science. You've got an institution around that, [84:19] and a whole bureaucracy acting as a proxy  for what we think that public good is, [84:25] with a whole song and dance around how to  make them correctly predict that your progress [84:31] will be in the spirit of that funding. Sometimes it's just straight-up teaching. [84:36] People like to send their kids to an  institute that has experts teaching them. [84:40] You provide brand value by being an expert,  and direct value by being a teacher. [84:47] Regardless of whether AIs are proving theorems or  not, or whether we're talking about 2016 or 2026, [84:52] that is something not enough students thinking  "I want to be a mathematician" consider. [84:56] I think it's worth thinking about. For me, I wasn't necessarily thinking about it, [85:02] and I stumbled into a career path where math  exploration can be monetized as entertainment. [85:08] I stumbled into that and I'm very  grateful I did, but it was an accident. [85:12] It wasn't deliberate. I could have avoided relying  on serendipity and done it a little bit more by [85:19] design had I been thinking critically about it. To your question—if we have almost-automated [85:27] theorem proving, and let's say they're  also really good explainers so you even [85:32] get the human understanding—I think a lot  of the social role that mathematicians [85:37] serve actually doesn't change that much. As a public, we still feel there's value [85:45] to basic science, and we trust  the judgment of mathematicians [85:48] to determine where their time is best spent. The prestige comes from within that community. [85:53] It's other members saying that a result  is really good, more than the grant [85:57] writer really understanding algebraic number  theory to understand it was a good result. [86:02] There's going to be an inner culture of  what constitutes valuable contributions. [86:06] Maybe it shifts away from theorem proving  and towards good definition writing. [86:10] Maybe it's that museum curator idea. But you're going to have that same [86:14] community as long as society as a whole is  still valuing the premise of basic science. [86:19] And if we're in the abundance world  that AI brings, there's probably more [86:24] funding in that direction in some sense. On the side of prestige to institutions [86:29] for who their lecturers are, I actually think  teaching is one of the most stable post-AGI [86:35] jobs that there is, because it's so relational. This is where parents want to spend their money [86:40] if they have an abundance of wealth:  on good teaching and good educating. [86:46] It goes so far beyond explanations. Even if LLMs are good explainers, [86:49] the thing that a teacher is doing is such  a social, coaching, mentor-type thing that [86:55] that's probably one of the most stable careers  that's going to exist over the next fifty years. [87:01] Insofar as a lot of mathematicians' roles  overlap with that, as the prospective student [87:07] going into it, you could lean into that. I actually think a lot more students should [87:11] think about and give credence to the idea of  being just a math educator and the value that [87:16] can serve towards the next generation. I'll couch again that I don't think I'm [87:22] the one to say, "Here, prospective young  mathematician, here's how you should think [87:27] about the future," because I'm a YouTuber. I'm not in the institution that they're [87:33] thinking of going into, so I'm  speaking as an outsider looking in. [87:36] But it feels like generally good, universal  advice: know where the money is coming from, [87:41] know where you plug into that. And if you're just asking those questions, [87:46] you're actually already steps ahead of all the  other fledgling prospective mathematicians. [87:51] In fact, think about the crazy world  where, within five or ten years, [87:56] the AIs are coming up with not only solutions  to the Millennium Prize problems, but totally [88:04] novel problems to be solving in the first place,  novel mathematical fields and objects and stuff. [88:09] It is in that world where, first  of all, there's a ton of abundance. [88:12] Two, the thing AI minds will have gone furthest  in, where they will have seen furthest beyond [88:20] our horizons, will be mathematics. There will be so much demand for, [88:26] "What have the AIs seen? Can you explain it to us?" [88:31] In that world, if there are any jobs  whatsoever, surely distilling what the [88:34] AIs have learned will be one of them. Also, it's funny because all of [88:39] this presumes that it's useless. We're not talking about the actual [88:42] practical applications of what math is being done. Insofar as there's any economic utility to it, [88:48] you would imagine that the people who  understand it and are able to make the [88:52] decision of where it should point actually have  a lot more economic value by being able to make [88:57] that judgment as a curator and point this  behemoth of new math in a useful direction. [89:03] Suddenly, that's a much more levered  move to make than it had been previously. [89:07] Can I ask you about that? Obviously, one question for AI for math [89:13] is not only can it do it, but is it any good? Or is it good for anything? [89:20] You were describing all the ways  in which, with group theory, [89:24] we're trying to figure out random facts about  the roots of different kinds of functions, [89:28] and now there are all these different applications  that are practical across many different fields. [89:33] Do you have some sense of whether, if we  just totally get to a place where the field [89:38] of human mathematics is accelerated 10X or  100X and some crazy shit happens, or are [89:45] we just going to be bottlenecked by other fields? I think there are some fields that probably will. [89:55] It's super spiky. With progress in  algebraic number theory, it feels [90:00] unlikely that that then unlocks something. But I remember talking to this mathematician [90:06] who does more dynamics and PDE-solving type stuff. He was referencing that his group had some ideas. [90:16] Let me see if I summarize this right. It’s like the way Boeing would make [90:19] planes is that they'd make it, do a bunch  of tests, and they had to disassemble it [90:24] and reassemble it based on those tests. His group essentially had some insights on [90:27] how to do more in simulation such that you  don't have to deconstruct and rebuild it. [90:32] It saved Boeing billions of dollars or something,  and then they just started funding that group. [90:37] That's much more obviously application-adjacent,  because PDEs just are that. [90:45] Progress in that domain, you would  imagine actually does unlock some things. [90:50] I don't know if it's these step changes,  but maybe it's more on the side of [90:55] engine design becoming a little bit more fluid,  or coming up with the right wing shape instead of [91:00] running a whole bunch of complicated CFD. Maybe you're able to speed up your CFD [91:04] simulations because certain pure math  insights make those more efficient. [91:09] I bet you'd just see a lot of great  incremental improvement there. [91:14] It seems less likely that the massive  breakthroughs in math immediately turn [91:20] into this massive economic breakthrough, like you  solve the Navier-Stokes problems, and then that [91:26] unlocks an ability to simulate more things. But you probably will see, at those fringes, [91:32] some meaningful leakage out of the  pure math insights into other things. [91:39] There's a ton of people working  on things like AI engineering, [91:43] physical engineering, and material science. You have to imagine they'd be in a good position [91:50] to look at the AI math insights and decide  whether they're relevant in some way or not. [91:57] It's another one of these things where I'm  not going to sit here and put a flag in [92:00] the sand predicting that there will be. But it'd be a little bit disappointing [92:03] and a little bit surprising if there  weren't, over the next five years, [92:07] economically valuable improvements made that were  directly referable to the AI progress in math. [92:15] It would just be disappointing if it was just  taking down a bunch of Erdős problems and [92:18] none of them were doing any of the math that  actually directly touches the physical world. [92:25] To your point about how a lot of the  history of mathematics was about building [92:28] up these piles of concepts and connections. Sometimes the piles connect with each other, [92:34] or you discover an application somewhere else. At the very least, [92:38] you just build up this huge pile. Then as broader progress in society [92:43] happens during the singularity, when we get  to the industrial part of the singularity, [92:47] you just have all these different ideas that  hopefully are useful in other parts of the world. [92:53] As I said, one of the interesting things  about what's happening is it causes [92:57] people to step back and ask, "What is math?" Maybe one of the awkward conclusions will be [93:02] revealing that it's just become wholly useless. The kind of questions being asked have become so [93:09] divorced from things that are physically  applicable that that's one of the things [93:12] mathematicians have to come to terms with. Everyone will look and say, "Hang on a second, [93:17] weren't you guys supposed to…  If there's 10X progress there, [93:20] why aren't we seeing it over here?" And then mathematicians are like, "Ugh." [93:23] Every time we wrote those grant proposals and  said, "Trust us, the elliptic curve progress is [93:28] going to help with cryptography," it shines  a light on the fact that maybe it doesn't. [93:32] So that's one possibility. Grant, this was super fun. [93:36] Thanks so much for doing it. Absolutely. My pleasure.