数学中的 AI 危机：Navier-Stokes、黎曼猜想与复杂性理论——Ankur Moitra、Steven Brunton 与 Alex Kontorovich 共谈数学的未来

好的，这是一期紧急播客。我认为可以公平地说，关于数学和人工智能的现状，这是数学史上的一个危机时刻。过去几周，机器声称解决了数学史上一些最难的问题：纳维-斯托克斯方程、准黎曼假设、唯一游戏猜想，以及数百个其他结果。所以，今天和我在一起的有 Steven Brunton，他是数学家、华盛顿大学的流体动力学专家，可以谈论纳维-斯托克斯、湍流、所有流体相关的问题，以及许多其他话题。还有 Ankur Moitra，麻省理工学院的数学家和理论计算机科学家，今年在普林斯顿高等研究院访问，可以谈论许多事情，包括唯一游戏猜想和复杂性理论，以及人工智能如何学习的数学和所有理论计算机科学。还有 Alex，罗格斯大学的数学家、数论专家，也可以谈论非常多的话题，包括黎曼假设和 Lean、自动证明和自动验证检查等等，以及数学中的许多其他话题。三位都很出色。他们都非常友善，这么早起来和我交谈。我组装了这个多智能体的小型人类大脑集群，试图解读发生了什么。在这个时刻，许多数学家，包括陶哲轩和其他杰出的数学家，都感到惊慌，混合着敬畏、恐惧、 dread，还有一丝希望的微光。所以，我很抱歉以一个关于感受的问题开始，但你对这个历史性时刻有什么感受？也许，Alex，你想先来？

All right, so this is a kind of emergency episode. I think it's fair to say about the state of mathematics and artificial intelligence, a kind of crisis moment in the history of mathematics. Over the last few weeks, machines claimed to have solved some of the hardest problems in the history of math: Navier-Stokes, the quasi Riemann hypothesis, the unique games conjecture, and hundreds of other results. So, here with me, we have Steven Brunton, who's a mathematician, fluid dynamics expert at University of Washington, who can speak to Navier-Stokes, turbulence, all things fluid, and many other topics. Also here is Ankur Moitra, mathematician and theoretical computer scientist at MIT, visiting the Institute for Advanced Studies in Princeton this year, who can speak to many things, unique games conjecture and complexity theory in general, and the mathematics of how the AI learns and all things theoretical computer science. And Alex, a mathematician, number theorist at Rutgers, who can speak to also very many topics including Riemann hypothesis and Lean, the automated proof and automated verifying and checking and so on and so many other topics in mathematics. All three are brilliant. All are very kind to show up this early morning and talk to me. I assembled this kind of multi-agent mini swarm of human brains to try to interpret what happened. In this very moment, a lot of mathematicians, brilliant mathematicians including Terence Tao and others are freaking out, some mix of awe, terror, dread, and few inklings of light of hope, also. So, I apologize to start with a question about feeling, but what have you been feeling about this historic moment? Maybe, Alex, do you want to go first?

是的，我不知道我会不会称之为危机。危机中有很多机会。这是一个重新思考我们试图实现什么的机会。正是机会的混乱，有些人对此反应消极，有些人反应积极，有时这些反应发生在同一个人身上。所以，我认为我们正在进入，如果不是已经进入的话，某种数学知识的黄金时代，在几千年来我们拿着一根蜡烛在树林里行走，只能看到这么远之后。现在，我们有了手电筒。

Yeah, I don't know if I will call it a crisis. There's a lot of opportunity in that crisis. It's a chance to rethink what it is we're trying to accomplish. And it's the chaos of opportunity that some people are reacting negatively to, some people are reacting positively to, and sometimes those reactions are happening in the same person. And so, I think we're entering, if we're not already in, a period of some kind of golden age of mathematical knowledge after millennia of having a candle that we are walking around the woods, and this is as far as we can see. Now, we have a flashlight.

我认为我们应该为不了解的人评论一下这个时刻的重要性。所以，也许引用你的话，类似‘天哪，如果人类做到这个，会立即获得菲尔兹奖。’也许你没说天哪，但基本上是说这是一个不可思议的结果。所以，很多人不会预料到在这么短的时间内取得这么大的进展。所以，这确实令人震惊。所以，你现在是作为一个在过去几天发现这件事后处理自己感受的人说话。所以，也许回到你第一次看到OpenAI发布的报告时的感受会很有用。你们俩碰巧同一天在镜头前，可能因为会议或聚会喝了几杯。也许，Ankur，你能说说你当时的感受吗？

I think we should comment on the magnitude of this moment for people who are not aware. So, maybe to quote you, something like, 'holy [expletive], if a human did this, it would be an instant Fields Medal.' Maybe you didn't say holy, but basically saying this is an incredible result. So, a lot of people wouldn't have expected this amount of progress in this amount of time. So, this is truly shocking. So, you're now speaking as a person who is dealing with the feelings you felt over the last couple of days having found this out. So, maybe also it'd be useful to go back to the feeling you were having when you first saw the report that in this case OpenAI dropped. And the two of you happened to be on camera loosely on the same day. Maybe a few drinks in because it was a conference or get-together. Maybe, Ankur, can you speak to what you were feeling?

结果发布的那天晚上很有趣。现在，这并不完全出乎意料，因为几周来一直有关于我所在领域两个最大问题的传言，它们就在那批结果中。所以，唯一游戏猜想和 RL 等于 L。甚至有关于矩阵乘法的传言。所以，我最初的反应是哇，都是真的。谢天谢地终于出来了，因为现在人们可以真正去研究它，处理发生了什么，并弄清楚下一步该做什么。现在，关于你之前的问题，我们感觉如何？嗯，对我来说，这每天都在变化，但我认为最首要的感受是真正感觉被拉向两个非常不同的方向。我的工作一直是关于提出正确的问题。从这个意义上说，这些工具对我来说很棒，因为它们增加了我学习其他领域的能力。就像有一个世界专家在那里用你自己的语言教你东西。你建立联系的能力被放大了。但是，你知道，我工作中我最喜欢的部分之一是指导方面。不仅仅是培训研究生和博士后，看着他们做伟大的事情，而是让他们和我一起踏上这些旅程。所以，把我拉向另一个方向的部分是认识到我们长久以来的成功剧本和蓝图现在被抛到窗外了。所以，建议学生和博士后应该做什么工作才能产生影响、成名，变得更加困难。所以，我感到非常兴奋，但也许在智力上比以前更孤独。部分原因是很多时候伟大的结果来自提出非常谦逊的问题。至少对我来说是这样。现在有太多的焦虑和恐惧，感觉哇，你必须做出真正重大的事情才能在 AI 结果的海洋中脱颖而出。感觉我们失去了一些耐心，去播下种子，让伟大的橡树生长。

It was an interesting night when the results dropped. Now, it was not entirely unexpected because for weeks there'd been rumors about the two biggest problems in my areas that were in that collection of results. So, the unique games conjecture and RL equals L. There are even rumors about matrix multiplication. So, really my first reaction was wow, it's all true. Thank goodness it's finally out there because now people can actually go through it and process what's happened and figure out what to do next. Now, as to your earlier question, you know, what are we feeling? Well, for me it changes on a daily basis, but I think the biggest most overarching feeling is really feeling like you're getting pulled in two very different directions. My work has always been about asking the right questions. And to that extent, these tools are amazing for me because they increase my ability to learn about other fields. It's like having a world expert who's there to teach you about things in your own language. Your ability to make connections is just magnified. But, you know, one of the parts of my job that I really love the most is the mentorship aspect. Not just training grad students and postdocs and seeing them do great things, but having them come with me on these journeys. So, the part that pulls me in the other direction is the recognition that the playbook and the blueprint for success that we had for so long is now thrown out the window. So, it becomes a lot harder to advise students and postdocs about what are things they should be working on that are going to make an impact, are going to make a name for themselves. So, I feel very excited, but maybe more intellectually lonely than I felt before. In part because a lot of times great results come from asking very humble questions. At least for me. And now there's so much anxiety and dread and feeling that wow, you would have to do something really momentous to stand out in the sea of AI results. That it feels like we've lost some of that patience in planting the seeds from which the great oak trees grow.

菲尔兹奖得主陶哲轩写道：‘我感到恐惧，许多人数年的耐心和刻意缓慢的研究努力，尤其是研究生和博士后，针对许多数学的激励性问题，被这样的发布随意扰乱或摧毁。就像一个人无法忘记电影剧透或填字游戏线索一样，一旦知道现有解决方案，就无法再那么有益和丰富地探索问题，我哀悼未走的道路和在开发技术的疯狂竞赛中失去的机会。’所以，我的意思是，如果现实一点，对博士生和博士后来说，这比你描述的更戏剧性。我是说，你说得相当平静，但如果我现在是博士生，我会处于震惊状态。就像，我一生都在经历这些。我知道很多朋友是程序员。去年十二月他们经历了这种震惊状态，基本上大多数编程世界转向 AI 编写大部分程序。所以，我会把它近似为一种失去意义的感觉，一种恐惧，黑暗的东西。我们会继续探索。Steve，你怎么想？你感觉如何？

Terence Tao, Fields Medal winner, wrote, 'I am horrified by the many person-years of ongoing patient and deliberately slow research efforts, particularly by graduate students and postdocs, toward many motivating problems of mathematics being casually disrupted or destroyed by such a release. Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution, and I mourn the path not taken and the opportunities lost in the frantic race to develop the technology.' So, I mean, if we're being realistic, it's more dramatic for a PhD student and a postdoc than maybe the way you've articulated. I mean, you're saying it quite calmly, but if I was a PhD student right now, I'd be in a state of shock. Like, my whole life I was going through some of this. I know a lot of my friends are programmers. They're going through this state of shock last December when basically most of the programming world switched to AI writing most of the programs. So, I would approximate it to a feeling of kind of a loss of meaning, kind of terror, like dark stuff. We'll keep exploring that. Steve, what do you think? What are you feeling?

我会说我的大部分感受是充满希望的。我认为我是一名工程师。我喜欢解决问题，使技术对人类更现实，比如人造心脏和飞机，而我们解决实际应用问题的能力将会变得好得多。

I would say most of my feelings are pretty hopeful. I think I'm an engineer. I like solving problems that make technology more realistic for humans, artificial hearts and airplanes, and our ability to solve real applied problems is going to get much much better.

我们的范围和影响将会扩大，因此我对此非常乐观。但存在很多不确定性，我认为其中一些对意义、对失去控制、对被排除在我们都热爱参与的过程之外的恐惧，是有点可怕的。而且，如果未来 5 到 10 年出现两种路径之一——我们真的被其他智能所 dwarf，或者它趋于平稳，成为我们用于好奇心、用于人类解决问题的不可思议的工具——我想我们谁都不会感到惊讶。这种不确定性是巨大的。那是两个截然不同的世界。所以不确定性对人类来说是困难的。我认为对我们这些选择数学的人来说也是困难的。我们选择数学，大概是因为我们喜欢规则和形式主义，而不确定性让人不舒服。不过，我认为谦逊的一面非常重要。这是一个很好的机会，让我们记住我们为什么这样做。比如，人类的目标是什么？不是出名。不是获得菲尔兹奖。解决有趣的问题，帮助他人，保持好奇，只是想理解世界。这些真的是很好的东西，值得回归。比如，我们需要什么才能成为满足的人类？我认为这是一个非常有趣的时刻，让我们思考。我最近一直在思考这个问题。对年轻人来说更困难，对吧？比如，如何找到自己的路，但从某种意义上说，我实际上对他们比对我们更有信心。

Our scope and reach is going to increase, and so for that I'm extremely hopeful. But there is a lot of uncertainty, and I think some of this fear of meaning, of losing control, of just being kind of cut out of the process that we all love being a part of, it is a little scary. And I don't think any of us would be surprised at either of the two paths that might take place over the next 5 or 10 years, where we really are just dwarfed by other intelligence, or it kind of levels off and this becomes an incredible tool that we use for our curiosity, for our human problem-solving. And that uncertainty is massive. Those are two very different worlds. And so uncertainty is difficult for humans. I think it's also difficult for those of us who have chosen math. We chose math probably because we liked rules and formalism, and uncertainty is uncomfortable. I think the humility aspect is really important, though. This is a really good opportunity for us to remember why we're doing this. Like, what is human purpose? It's not to be famous. It's not to have a Fields Medal. Solving interesting problems, helping other people, being curious, just wanting to understand the world. Those are really good things to kind of get back. Like, what do we need to be satisfied humans? And I think it's a really interesting moment for us to think about. I've been thinking about it a lot lately. It is more difficult for young people, right? Figuring out how to find their way, but I actually have more faith in them than in us in a sense.

你认为 Terry 说的有道理吗？当你知道一个问题已经被解决时，就更难有动力去探索那个空间，尤其是当问题被 AI 系统这样解决时？

Do you think there's something to what Terry says about it's harder to be motivated when you know a problem has been solved? To explore that space when a problem has been solved like this by an AI system?

是也不是。从某种意义上说，我非常同意路径和目的地一样重要。就像如果你去公路旅行，公路旅行通常和到达那里一样有趣。同时，如果我们发现另一个国家发明了传送或时间旅行，我保证我们会在未来 5 年内重新发明它。就像如果你知道这是可能的，并且你可以稍微逆向工程一下，我们会好奇，我们会深入其中，我们会弄明白。所以，知道有解决方案也是更快到达那里的巨大动力。

Yes and no. In the sense that I do very much agree that the path is as important as the destination. Like if you go on a road trip, the road trip is usually just as fun as getting there. At the same time, if we found out that another country had invented teleportation or time travel, I guarantee we would reinvent it in the next 5 years. Like if you knew it was possible and you could reverse engineer a little bit, we would be curious, we would dive into it, and we would figure it out. So, knowing there's a solution also is an incredible motivation to get there much faster.

让我这样问这个问题。如果这一刻标志着 AI 基本上，引用一下，解决数学或杀死数学的轨迹时刻，你能论证这确实是一个轨迹吗？然后你能论证这不是一个轨迹吗？你能对两条路径都进行钢人论证吗？这是一条通往我们所知的数学完全被完成的路径。所以，那将成为机器做的事情。然后人类做的是根本不同的事情。或者仍然会有类似于我们当前数学的东西。你能对两种情况都进行钢人论证吗？

Let me ask the question this way. If this moment marks the moment of the trajectory where AI basically, quote unquote, solves math or kills math, can you make the case that that is in fact a trajectory? And then can you make the case that that is not a trajectory? Can you steelman both paths? That this is a path towards which mathematics as we know it is done completely. So, that's going to be a thing done by machines. And then what humans do is something fundamentally different. Or there still will be something that looks similar to our current mathematics. Can you steelman both cases?

是的，你说了两件事，我想把它们分开。一个是解决数学，另一个是摧毁数学。所以，让我们把解决说得精确些。解决数学是什么意思？如果你回想国际象棋引擎刚出现的时候，人们说：“哦，这会很棒。我们将下半人马棋。”所以，这是人类加计算机系统对抗另一个人和计算机系统。这持续了大约 10 秒钟，然后计算机变得如此之好，以至于如果你不做任何事，除了选择计算机推荐的最佳着法，你就会输。所以，人类在那游戏中的角色变成了，是的，接受第一步。是的，接受第一步。那不再有趣了。我们再也看不到多少半人马棋了，对吧？所以，事实上，我刚刚和一位名叫 Thomas Gobert 的同事进行了一次对话，我们将我们对未来可能样子的论点公理化。我愿意为了论证而接受的一个公理是我称之为“已解决公理”的东西。已解决公理说，让我们假设在某个未来时刻——我们现在远未到达那里，我也不期望我们一定会到达那里；我们会看到能力是否会在某处逐渐减弱。但让我们假设已解决公理，它说任何可以通过人类与 AI 互动解决的数学问题，也可以由 AI 单独解决。这就是已解决公理。所以那意味着，真的，人类没有什么新的事情可做——你可以对 AI 系统说，好吧，我有问题 X。我想思考问题 X，但我取得进展的唯一方式，我在问题 X 上取得的任何进展都只是按下按钮。AI 思考它。它不一定解决它。这并不意味着它会解决每一个问题。黎曼猜想还没有解决。好吧？那才是我真正想要的解决方案。我只是没想到在我有生之年会看到我们几天前看到的任何进展。所以，并不是 AI 系统在解决每一个问题。它们在解决每一个可以通过人类与 AI 互动解决的问题。这就是那个公理的意思。所以我说，如果我们曾经在那个世界里，那必然会摧毁数学吗？我强烈的论点是，不。我不认为它会摧毁数学。

Yeah, so, you said two things and I'd like to separate them. One is to solve math and the other is destroy math. So, let's make solve precise. What does solve math mean? If you think back to when chess engines first came out and people were like, "Oh, this is going to be amazing. We'll play centaur games." So, this is a human being plus computer system against another human being and a computer system. And that lasted like 10 seconds and then the computers got so good that if you did anything but choose the top move that the computer recommended, you would lose. And so the human being's role in that game became, yes, accept the first move. Yes, accept the first move. That's no longer fun. And we don't see many games of centaur chess anymore, right? And so, in fact, I just had a conversation with a colleague named Thomas Gobert where we axiomatized our argument for what the future might look like. And one of the axioms that I'm willing to even accept for the sake of argument is what I call the solved axiom. And the solved axiom says, let's assume at some future point—we're nowhere near there now, and I don't expect us to be there necessarily; we'll see where the capabilities taper off if they do. But let's assume the solved axiom, which says any mathematical problem that can be solved with human interacting with AI could also have been solved by AI alone. That's the solved axiom. So that would mean that really there's nothing new for a human being to—you can say to an AI system, okay, I have problem X. I want to think about problem X, but the only way that I will make progress, any progress that I'll make on problem X will just be press the button. The AI thinks about it. It doesn't necessarily solve it. It doesn't mean that it'll solve every problem. The Riemann hypothesis is not solved. Okay? That's what I really want the solution to. I just didn't think in my lifetime I would see any kind of progress towards it that we saw a few days ago. So, it's not that AI systems are solving every problem. They're solving every problem that could be solved by humans interacting with AI. That's what that axiom would be. So I'm saying, if we are ever in that world, does that necessarily destroy mathematics? And my strong argument is no. I don't think it destroys mathematics.

如果那个公理是真的，人类的角色是什么？

If that axiom is true, what's the role of humans?

所以，我不认为数学，纯数学，不是工程数学，不是那些能治愈癌症和解决喷气机——每个人都拥有喷气背包或你想象的其他技术进步。纯数学，我们知道我们证明定理只是因为我们好奇。我不认为定理的解决方案以 1 和 0 的形式存在于数据中心对人类有价值。我认为纯数学的价值在于它活在人类的头脑中。

So, I don't think that mathematics, pure mathematics, not engineering mathematics, not things that will cure cancer and solve jet—everybody has a jetpack or whatever other technological advancement you might imagine. Pure mathematics, we know that we're proving theorems just because we're curious about them. I don't think the solution to theorems living in ones and zeros in data centers is valuable to human beings. I think the value of pure mathematics is that it lives in the heads of human beings.

所以，补充一下，我很高兴你直接提出了每个人心中的问题，谢谢。我要提醒不要过度拟合于一个目标函数，对吧？因为我们曾经认为自己是周围最聪明的东西，人类数学家可以解决，达到最高的树梢，但现在我们面临的事实是，许多我们认为还很遥远的问题已经被 AI 解决了。但也许我们可以得到的安慰之一是，与 AI 解决它的方式相比，我们在计算上非常有限。给定证明同一定理的相同终点线，我无法同时记住那么多论文或想法。

So, just to add to this, I'm glad you came right out with the question that's on everyone's mind, so thank you for that. I would caution against overfitting to one objective function for this, right? Because we used to think of ourselves as just the smartest things around, human mathematicians can solve, reach the highest treetops, but now we're faced with the fact that many of the problems that we thought were quite a ways out have been solved by AI. But maybe one of the consolations we can take is actually the fact that we're very computationally limited compared to the way that AI is solving it. Given the same finish line of proving the same theorem, I can't really hold that many papers or ideas in my head at one time.

所以，为了达到同样的终点，也许我需要开发更通用、更可抽象的方法，这些方法不仅能帮你解决那一个问题，还能解释联系或你想进入的其他方向，对吧？所以，当你回顾自己写的论文时，总是很神奇，你能看到这些有时看不见的线索，它们连接不同领域的工作，因为开发理解问题的系统方法有助于你解决周围的问题，而不仅仅是得到一个简洁形式化的定理。

So, in order for me to reach that same finish line, maybe I have to develop more generalizable and abstractable ways of getting there that then help you not just solve that one problem, but explain connections or other directions you want to get into, right? So, it's always amazing when you look back on papers you've written and you can see these threads, sometimes invisible, that connect different areas of work because developing systems of ways of understanding problems helps you for surrounding problems, not just to get a lean formalized one theorem down.

但有趣的是，你说的那些有时看不见的线索，现在对 AI 系统来说却相当可见。似乎这正是它们能取得如此大进展的真正原因，因为它们能提取线索并加以利用，因为它们能看到这么多论文。

But the interesting thing here is those threads, what you're saying is sometimes invisible, are quite visible to the AI systems now. It seems like that's perhaps the actual very reason why they're able to make so much progress is they can extract the threads and pull on them because they have the view of so many papers.

我认为这取决于你所说的线索是什么意思。对我来说，证明结果的原因是为了更好地理解你周围的世界。数学对我来说是抽象和理解事物的工具。当你有一个重大结果时，重要的不仅仅是陈述的真实性，还有你在完成过程中学到的东西。通常人们的第一个问题是，‘好吧，你证明了一些奇妙的东西，但你是怎么做到的？’然后人们一旦消化了那个新想法、新定义、新看待事物的方式，附近就有很多其他问题迎刃而解。所以不知何故，你建立的结构不仅帮助你攻击一个问题，而是整个问题海洋。而且我们并不需要完全形式化的结果来做到这一点。有时我们用非常粗糙的比喻来推理，这些比喻帮助我们概念化周围的世界。

I think it depends on what you mean by threads. For me, the reason you prove results is so that you better understand the world around you. Mathematics to me is a vehicle for abstraction and understanding things. And when you have a major result, what's important is not just the truth of the statement but what you've learned on route to doing that. Usually people's first question is, 'Okay, you proved something fantastic, but how did you do it?' And then people, once they digest that new idea, that new definition, that new way of looking at things, there are lots of other problems nearby that fall. So somehow you're building structures that help you attack not just one problem, but a whole sea of problems. And it's not that we need fully formalized results to do that. Sometimes we reason in very sloppy metaphors that help us conceptualize the world around us.

这其中有多少是 Alex 离线提到的‘注意力就是你所需要的一切’？有多少是人类能够照亮那些所谓重要的领域的能力？也就是人类提出正确问题、重要问题的能力。这一直都是如此。

And how much of this is something Alex mentioned offline, attention is all you need. How much of this is the human ability to shine a light on particular areas that quote unquote matter. So the human ability to ask the right questions, the important questions. That's always been the case.

在 AI 出现之前，你可以这样想：这里有公理，仅仅从公理抽象地就可以得到逻辑推论，对吧？我们可以从公理证明一些定理。而我们可以证明的定理空间比人类真正关心的空间要大得多得多。所以几乎根据定义，这是一个同义反复：我们会为之兴奋的定理就是我们作为人类关心的定理。所以也许 AI 能说服我们存在一些有趣的结构。也许它可以说，‘嘿，看这里。让我给你这个新定义之类的。’它还没有做到这一点。

So within this before AI, there's a you can think of like here's the axioms and just abstractly from the axioms you can get to logical deductions, right? There are theorems that we could prove from the axioms. And there's a much, much, much larger space of theorems that we could prove than the space that human beings ever actually care about. And so by almost by definition, it's like a tautology that the theorems that we'll get excited about are the theorems that we care about as humans. So maybe AI can convince us that there's some interesting structure. Maybe it can say, 'Hey, look over here. Let me give you this new definition, or something.' It hasn't done that yet.

有时我们选择关心的事物的机制，它可能有应用。所以，纳维-斯托克斯方程与所有自然系统中的流体有关，但它也可能是一种美与困难的混合体，吸引着我们。也许有一种直觉。素数充满了这些非常复杂的问题，也许在直接意义上没有明显的应用，但不知何故吸引着我们。也许作为追求困难事物的副产品，伟大的应用数学和工程以及现实世界问题的解决方案就出来了。

And sometimes the mechanism by which we select the things that we care about, it could have an application. So, Navier-Stokes is connected to fluids in all natural systems, but it could also be a thing that some kind of mixture of beauty and difficulty that just pulls at us. Maybe there's an intuition. The prime numbers are full of these really complicated problems that maybe don't have an obvious application in the immediate sense, but it pulls at us somehow. And maybe as a symptom of the pursuit of the hard thing, great applied mathematics and engineering and solutions to real-world problems comes out.

举个例子，对吧？很多时候当我研究一个问题时，我可能有一个更大的目标想要实现，但通常我做的是尝试找到最简单、最令人尴尬的未解决问题，我仍然不知道如何解决。那种把它消化成‘天哪，我们怎么会不知道这个？’的方面，感觉是在正确的轨道上，因为然后当我们能解决那个例子时，也许我们正在组装我们需要攻击其他问题的工具集。所以，我确实认为我们将不得不变得更加有意识地思考我们关心什么？从元认知的角度，我们如何使用这些工具？我们到底委托给它们什么，不委托给它们什么？我们真的处于起步阶段，我们只是边走边拼凑。

So, just as an example, right? A lot of times when I'm working on a problem, I might have some bigger goal that I want to accomplish, but usually what I do is I try to find the simplest, most embarrassing open problem that I still don't know how to solve. That aspect of digesting it down to something, 'Boy, how can we not know this?' It feels like it's on the right track, because then when we can solve that example, maybe we're assembling the set of tools we need to make attacks on other problems. So, I do think we're going to have to become a lot more deliberate about what do we care about? How do we use these tools from a metacognition standpoint? What exactly are we delegating to them and what are we not delegating to them? And we're really at the beginning stages, where we're just trying to piece it together as we go along.

一方面，寻找证明有点像探索这棵可能性之树。你可能基于你见过的其他证明有一些直觉，知道该往哪个方向走。也许我应该走这个分支。也许我应该走那个分支。但有时我们遇到树中分支因子非常高的点。我们就是不知道往哪走。我们甚至没有猜测如果我们走下去会发现什么。现在在这个半人马世界的数学中，我们突然有了其他智能可以互动。而如何使用那个工具来寻找美丽的理论，我认为将是具有挑战性和不断发展的，但重要的是不要忽视我们不仅仅关心定理这一事实。

On the one hand, looking for a proof is kind of like exploring this tree of possibilities. You might have some intuitions based on some other proofs you've seen about which directions to go in. Maybe I should go down this branch. Maybe I should go down this branch. But sometimes we encounter points in the tree which have really high branching factor. We just don't know which way to go down. We don't even have a guess about if we went down them, what would we discover. And now in this centaur world of math, we suddenly have something else intelligent that we can interact with. And what is the right way to use that tool to search for the beautiful theories, I think will be challenging and evolving, but it's important to not lose sight of the fact that we don't just care about theorems.

所以，我是一名数学家，一名理论计算机科学家，但我也一直在更广泛地阅读社会科学，只是为了接受所有正在发生的变化。有一件事真正印在我脑海里，我很喜欢，叫做门卫谬误，非常有趣。所以，这个想法是你把一项复杂的工作归结为它最简单、最可见的任务。那么，门卫做什么？我猜他们在人们进出时帮忙开门，对吧？当我们住在纽约时，我们真的很喜欢和我们的门卫聊天。他们是很好的人，但如果你把它简化成那个简单的工作，你会想，‘哦，我可以用一个感应人们进出的滑动门来代替他们。’但你错过了他们工作中增加很多价值且难以表达的所有方面。我认为这也是思考 AI 时代数学的好方法。数学家做什么？工作中最简单、最可见的部分是我们证明定理。而关于数百年来未解决的猜想的定理，比如费马大定理，这些是非常可见的东西，我们花了很多时间普及它们，但我不认为这就是数学家的全部。我喜欢这样想：数学家是深入思考的人，深入思考的副产品是证明旧猜想，但也是建立新理论，建立看待旧事物的新方式，建立新的概念联系。

So, I'm a mathematician, I'm a theoretical computer scientist, but I've also been reading a little bit more broadly about social sciences just to sort of make my peace with all of the changes that have been happening. One of the things that really stuck in my mind that I liked is there's something called the doorman fallacy, which is very interesting. So, the idea is that you take a complex job and you boil it down to its simplest, most visible task. So, what does a doorman do? I guess they hold the door for people as they're coming and going, right? And when we lived in New York, we really enjoyed talking to our doorman. They were wonderful people, but if you digest it down to just that simple job, you can think, 'Oh, I can replace them just by a sliding door that detects when people are going or coming.' And you miss out on all the aspects of their job that add a lot of value and are harder to articulate. And I think that's a good way to think about mathematics in the age of AI, too. What does a mathematician do? The simplest and most visible part of the job is we prove theorems. And theorems about conjectures that were open for hundreds of years, like Fermat's Last Theorem, these are very visible things that we spend a lot of time popularizing, but I don't think that that's all that a mathematician does. I like to think about it as a mathematician is someone who thinks deeply, and a byproduct of thinking deeply is proving old conjectures, but it's also building new theories, building new ways of looking at old things, making new conceptual connections.

而这些部分，你知道，也许 AI 也会吞掉。但我们现在还没到那一步。所以，尽管这是一个重大时刻，我不认为现在是按下恐慌按钮的时候。我认为现在是真正重新聚焦我们价值观的时候，并作为一个领域进行这些对话。记住，我们所做的事情确实有多元的美学和目标。当一件事变得廉价，比如解决定义明确的问题，它会让与之正交的其他事情变得更有价值。

And these are parts that, you know, maybe AI will swallow up, too. But we're not there yet at this moment. So, even though this is a momentous time, I don't think it's the right time to hit the panic button. I think it's the right time to really recenter in terms of what our values are and have these conversations as a field. And remember that what we do really has a plurality of different aesthetics and goals. And that when one thing becomes cheap, like problem-solving for well-defined things, it makes the other things that are orthogonal to it all that much more valuable.

这真的很美。门卫谬误，去困惑，这里是指将一个人的工作简化为最明显的任务，并假设自动化那个任务就能取代整个角色的价值。一家酒店决定，引用，“我们付钱让人开门，而自动门可以更便宜地做到。”但事实上，门卫还欢迎客人、识别回头客、处理行李和叫出租车、提供安全并阻止不受欢迎的访客，并传递声望，让客人感到被关心。

That is really beautiful. The doorman fallacy going to perplexity here is the mistake of reducing a person's job to its most obvious task and assuming that automating that task replaces the value of the whole role. A hotel decides, quote, "We're paying someone to open the door, and automatic door can do that more cheaply." But, in fact, the doorman also welcomes guests and recognizes returning customers, handles luggage and hails taxis, provides security and discourages unwanted visitors, and signals prestige and makes guests feel cared for.

那是我最喜欢的部分。我查了门卫做什么，天哪，我们和数学家有很多共同点。如果我不传递品牌价值，我不知道我在做什么。

That was my favorite part. I looked up what a doorman does, and boy, we have a lot of commonalities with mathematicians. If I don't signal brand value, I don't know what I'm doing.

这很有趣。但我认为我们人类总体上，以及数学家的任务，是列出那个清单，真正更明确地说明人类到底做什么。我还没听到你们任何人提到美。如果我提到美太多，我道歉。但从我外行粉丝的角度看，数学家在某种意义上是阐释者，是广义上美好事物的学者。

That's funny. But I I I think the task for us humans in general, but also mathematicians, is to come up with that list, like actually make more explicit exactly what it is that humans do. And I haven't heard any of you mention, uh, beauty. I apologize if I've mentioned beauty excessively already. Uh, but there's something that mathematicians are, from my outsider sort of fan perspective, are kind of expositors, scholars of of of of things beautiful, broadly defined.

完全正确，直到 5 秒前，解决新的深奥定理的能力和我们作为人类数学家真正想要的东西之间几乎完美相关。而我们想要的正是你所说的，美、理解和思想的深度。而事实上，你可以按一个按钮，毫不费力、没有深入思考、没有必要的知识，就得到过去是我们的代理的东西。就像我们有人会说，做出突破，然后把他们发现的东西放到 Twitter 上，现在，数学家该怎么办？

That's exactly right is uh until 5 seconds ago, there was a near perfect correlation between the ability to solve new deep theorems and what we actually want as human mathematicians. And uh and what we want is exactly what you said, beauty and understanding and depth of thought. And uh the fact that you can press a button and get what used to be our our proxy uh without any effort, without any deep thought, and without any uh necessary knowledge. Like we have uh people saying, you know, do a breakthrough and then putting whatever they find on on Twitter and uh and now, you know, what are mathematicians supposed to do with that?

我会退一步说，在回答问题和解决问题之前，是提出正确的问题。所以，我认为衡量好的问题解决能力的方式是提出正确的子问题，提出一个有趣、可解决且可解释的问题。因此，我觉得提出好问题的价值一直是我们看重同事的品质，而且这会变得越来越有价值。

I would go one step behind answering questions and doing problem-solving is asking the right questions. So, like that is how I think you measure good problem-solving is asking the right sub-questions, asking, you know, a question that is interesting and can be solved and can be interpreted. And so, I feel like the value of of asking good questions is has always been something we value in our colleagues, and that's going to become increasingly more valuable.

我完全同意。并且，强调一个隐藏在那里的点，我认为提出正确的问题需要大量的数学洞察力。

I agree completely. And just to, you know, emphasize a point that was sort of um hidden in there, I think asking the right questions requires a lot of mathematical insight.

嗯。

Mhm.

绝对。

Absolutely.

你会看到人们一直评论，你知道，我们未来会重视什么？是提出正确的问题，是做好演讲。但你知道，我不会贬低，不是说你贬低了，但明确说，我不会贬低数学洞察力在这两项任务中的作用。因为当我们说演讲会变得更重要时，我并不是想象我们都会成为 Twitter 影响者或 TikTok 什么的。

You'll see people make that comment all the time, you know, what are we going to value going forward? It's asking the right question, it's giving great talks. But, you know, I wouldn't minimize, not that you were, but just to be explicit, I would not minimize the role in mathematical insight in both of those tasks. Because it's when we say that, you know, talks will become more important, I'm not imagining that, you know, we're all going to become Twitter influencers and Tik Tok whatevers.

不是。

Not.

我希望不是。因为真正突出的是拥有被你的社区采纳的想法，这些想法最终会随着时间的推移变得强大。所以，我想把这和 Lex 之前提到的事情联系起来，他问，我是不是过于乐观了？可能是吧？与研究生看待这些事情的方式相比。但我认为短期内我最担心的事情之一是我们的系统和机构会发生什么。你知道，不仅仅是招聘和终身教职等等。而是当研究生开始他们的博士生涯时，他们是领域的新进入者。他们试图吸收领域内的技术、重要问题，并试图弄清楚如何成名。我们过去会给他们一个非常简单的蓝图。拿一个人们关心的问题并改进它。然后人们就会知道你是谁，你会在顶级期刊上发表论文，门会为你打开。但现在我认为那个蓝图在很多方面都改变了，因为现在有太多结果发生，我们都超载了。我认为花时间重新思考那个蓝图是可以的。我们仍然关心同样的事情，比如影响你社区的想法，但它们更难用我们可以给他们的简单标准来操作化。所以我仍然乐观。我只是认为需要重新设计我们的许多系统和机构来做到这一点。

I hope not. Um because, you know, really what stands out is uh having ideas that get adopted by your community that end up becoming powerful over time. So, you know, I want to tie this back to something that Lex brought up earlier on, which is, you know, he asked am I being overly optimistic? Probably, right? Compared to uh you know, how grad students view these things. But, I think one of the things that I'm most worried about in the near term is what's going to happen to our systems and institutions. You know, it's not just hiring and tenure and all these things. But, it's really that when a grad student starts off their PhD, you know, they're a new entrant to the field. They're trying to absorb what are the techniques in their field, what are the important problems, and they're trying to figure out how to make a name for themselves. And we used to do is we would give them a blueprint that was very simple to follow. Take a problem people care about and improve it. And then people will know who you are, you know, you'll get papers in top venues, you know, doors will open for you. But, now I think that blueprint has changed in a lot of ways, because there are so many results that are happening right now that we're all just overloaded with it. And I think it's okay to take time to rethink that blueprint. We still care about the same things like ideas that are going to influence your community, but they're just harder to operationalize in terms of some simple rubric that we can give them. So, I'm still optimistic. I just think it's going to take, you know, reworking a lot of our systems and institutions to do that.

顺便说一句，很抱歉回到你的观点，我实际上不相信那个坚实公理。我不认为我们到了那一步。我可以证明一个定理，但我不理解为什么证明有效。这里发生了一些魔法，直到我以某种方式推广这个现象并更深入地理解它，我才会明白为什么证明如此有效。也许这是我不相信那个坚实公理的原因之一，因为我们做的事情更具探索性，什么构成一个好问题，不仅仅是“这是黎曼猜想，取得一些进展”。

By the way, then so sorry, to to go back to your point, I don't actually believe the solid axiom. I don't think we're there. I can prove a theorem, but I don't understand why the proof works. There's some magic that happens over here, and until I generalize somehow the phenomenon and understand it more deeply, then I'll see why the the proof works the way it does. Maybe that's one of the reasons why I don't actually believe the solid axiom, because the things that we do are much more exploratory and what constitutes a good question beyond just here's the Riemann hypothesis, make some progress on it.

所以，我认为你也提出了一个非常重要的点，即我们概念的普遍性和泛化。所以，与数学中的美相关，我认为黄金标准之一一直是你的结果能有多简单，以及它能泛化多少？你知道，在可证明的无限定理空间中，找到一些简单且可泛化的结构，我认为这仍然是黄金标准。除非我弄错了，人类仍然在提问。所以，我们用 AI 证明了纳维-斯托克斯方程的存在唯一性，但人类在引导这个。人类仍然在提问。比如，如果有一台机器能证明你提出的任何定理，我想看到下一个突破，我想让它开始告诉我应该问什么问题。

So, I think you also brought up a really important point this like generality and generalization of our concepts. So, related to beauty in mathematics, I think one of the gold standards has always been how simple can your result be and how much does it generalize? And you know, out of this infinite space of theorems that could be proven, uh finding some some structure that is simple and generalizes, like that is I think still the gold standard. And unless I'm mistaken, humans are still asking the questions. So, like we, you know, we prove uh existence uniqueness of Navier-Stokes with AI, but humans are guiding this. Humans are still asking the questions. Like I would love to see the next breakthrough if you had uh a machine that can uh prove any theorem you pose, I want it to start telling me what questions I should be asking.

你认为那台机器可能只是做一个 for 循环，然后找到所有简单且可泛化的，基本上所有 E=mc² 吗？

Do you think it's possible that that machine just does a for loop and uh finds all the simple and generalizable, basically all the E equals MC squared?

实际上，这正在发生。

That's happening today actually.

所以，确实有一些研究项目在系统性地、自动地发现最简物理定律。所以这不太是数学定理，更多是物理——支配性的定律和方程。

So, there are absolutely research programs that are systematically, automatically discovering minimal physical laws. So this is less mathematical theorems, more physics — governing laws and equations.

也就是观察自然系统，试图看看这些自然系统里是否存在简单的物理定律，能建模它们如何运作。

So, observing natural systems and trying to see in those natural systems if there are simple physical laws that model how they operate.

对，把它提炼到最基础的部分。

Yeah, distilling it down to the basics.

这样你就能重现已知的理论。如果你观察球下落，你能推导出引力和 F 等于 MA。如果你观察流体，你能推导出纳维-斯托克斯方程。然后把这套方法用到我们还没写下简单方程的数据上——天气、湍流、太阳耀斑之类的东西。我猜数学里也在发生类似的事，那里大概也有这些可以被推导出来的最简片段。

So you can kind of recapitulate known theories. If you observe balls dropping, you can deduce gravity and F equals MA. If you observe fluids, you can deduce Navier-Stokes. And then taking that to data where we haven't written down simple equations — weather, turbulence, solar flares, things like that. And I'm assuming similar things are happening in math, where there are probably these minimal pieces that can be deduced.

你觉得数学里是否存在一个很大的空间，里面都是简单且可推广的东西，可以被自动发现？

Do you think there's a large space of simple and generalizable things in math that could be discovered automatically?

所以，我对“已解决公理”的理解是：我作为人类数学家做的任何事——我好几次坐下来想，“好吧，LLM 永远做不了我做的这件事。”然后我就想，“嗯，为什么不能？”我可以问一个问题。我可以问它一个问题。我可以问它，“我应该问哪些问题？”

So, my version of the solved axiom is that anything that I do as a human mathematician — I've tried on several occasions to sit down and say, "Okay, LLMs will never do this that I do." And I'm like, "Well, why not?" I can ask a question. I can ask it a question. I can ask it, "What questions should I ask?"

嗯。

Mhm.

对吧？因为它是一个如此通用的系统——我不是说它现在在所有那些事上都擅长，对吧？但我对“已解决公理”的理解是：我作为人类能贡献的任何东西，它也能靠自己贡献出来。所以我真正想说的是，从价值角度看，我给这个社群带来什么？不能是“我问了一个好问题”。不一定能是——在这个“已解决公理”的世界里，再说一遍，我并不相信它——但即便在那个世界里，也应该是我问了一个好问题、我证明了一个好定理之外的东西。而且即便在那个世界里，我认为你仍然需要数学家，需要具备人类数学专长的人。

Right? Because it's such a generalized system — I'm not saying that it's good at all of those things now, right? But my version of the solved axiom is anything that I can contribute as a human being, it could also have contributed on its own. And so, what I really mean by that is in terms of value, what do I bring to the community? It can't be "I asked a great question." It can't necessarily be — in this world of the solved axiom, which again, I don't believe — but even in that world, it should be something other than "I asked a great question, I proved a great theorem." And even in that world, I think you need mathematicians, people with human mathematical expertise.

好，你相信“已解决公理”吗？在我们能多接近它的这个谱系上，你觉得我们在哪儿？在这个谱系里，你认为人类的角色是什么？

Okay, do you believe in the solved axiom? Where on the spectrum of how close we can get to it? And in that spectrum, what's the role of the human, do you think?

我想我并不真的相信“已解决公理”。我觉得它在某些方面也许对了一半，而弄清楚哪一半是对的，会非常有趣也非常有挑战。所以我认为在 AI 能解决什么这件事上，还会有重大惊喜和更多进展。但对我来说，当你面对一项非常非常强的新技术时，危险在于：当你试图去概念化它为什么有效时，你会通过拟人化来理解它。所以这里较早的一个教训也许来自深度学习。有一些很惊人的例子——你可以用神经网络做到人类水平甚至更好的物体识别。但总体而言，它成功的方式不如人类那样稳健。因为存在对抗样本这类例子：你可以给自然图像加上非常细微的图案，就能完全骗过物体检测系统，却骗不过人。我在 MIT 有位前同事，Aleksander Madry。他们做过一些很好的工作，我很喜欢。我记得标题大概是《对抗样本是特征，不是缺陷》。他们做了下面这个很酷的实验。你拿一堆自然数据，也许是猫和狗的图片。你训练一个深度网络把它们识别得很好。你给它们加上一些对抗扰动，让——在 AI 系统看来——猫看起来像狗，反之亦然。所以它就像一个小小的图案，改变了物体检测器的工作方式。然后你造一个完全虚假的数据集。里面有猫的图片，上面叠着狗的图案，你把它标成狗，猫则反过来。神奇的是，当你用这些全错的标签训练一个深度网络时，它会学到在自然数据上表现很好的东西。所以，不知怎么的，当我们识别猫和狗时，我们是以一种能用腿、尾巴和整体形状来表述的方式去做的。但很多时候，当深度网络或 AI 能做事时，它们可能会找到解决问题的暗门或后门，利用数据中大量人类不用的统计关系。所以对我来说，重要的是不要在你所能做的和 AI 不能做的之间划出非常鲜明的界限，因为你会出错。但这并不意味着我不相信会出现某种界限，哪怕只是因为我认为我们过度拟人化了它们解决问题的方式。而当我翻看 OpenAI 的解答时，它们看起来就不像是人类解出来的。所以对我来说，似乎仍然存在非常大的质的差异，而这其实是合作和半人马数学的机会，而不是对 AI 霸主的恐惧。不过，谁知道呢？下周可能就错了。

I guess I don't really believe in the solved axiom. I think maybe it's half true in some ways, and figuring out which half of it is true will be very interesting and challenging. So I think there'll be major surprises and more to come in terms of what AI can solve. But for me, when you're facing a new technology that's really, really good, the danger is that when you try to conceptualize why it's working, you do so by anthropomorphizing it. So maybe one of the earlier lessons here is from deep learning. So there are these amazing examples — you can take neural nets and you can do human-level or better object recognition. But in general, the way that it succeeds is less robust than how a human does. Because there are all these examples like adversarial examples, where you can add very slight patterns to natural images that can completely fool the object detection systems, but not a human. And I had a former colleague at MIT, Aleksander Madry. They had some really nice work, which I really enjoyed. I think the title was something like "Adversarial Examples Are Features, Not Bugs." So they did the following cool experiment. You take a bunch of natural data, maybe pictures of cats and dogs. You train a deep network to identify them very well. You add some adversarial perturbations to them that make — to the AI system — a cat look like a dog and vice versa. So it's kind of like a small little pattern that changes how the object detector works. And then you create a totally fake data set. It's got pictures of cats with dog patterns on top, and you label it as dog, and vice versa for cats. And amazingly, when you train a deep network on this with all the wrong labels, it's going to learn something that does well on natural data. So somehow when we recognize what a cat and dog is, we do it in a way that we can articulate in terms of its legs and its tail and its shape in general. But a lot of times when deep networks or AI can do things, they might be able to find trapdoors or backdoors to solving the problem that use a lot of statistical relationships in the data that a human doesn't. So to me, it's important to not draw this boundary very sharply between what you can do and what AI can't do, because you're going to be wrong. But that doesn't mean that I don't believe there will emerge some kind of boundary, if only for the fact that I think we over-anthropomorphize the way that they solve problems. And when I'm looking through the OpenAI solutions, they just don't look like a human has solved those problems. So to me, there still seems to be a very big qualitative difference, and that's an opportunity really for collaboration and centaur math more than it is dread about the AI overlords. But who knows? Could be wrong next week.

你的直觉是，这其中有多少是 AI 只是把成千上万篇不同的论文串起来、顺着线索去拉——抱歉又用回这个比喻——但就是不同论文的线索，然后基本上只是把不同论文里已经存在的证明表述出来，做一些小的连接？还是它真的在发展新的数学思想？你有没有看到那种天才般的 AlphaGo 围棋“第 37 手”的微光？

What's your intuition about how much of this is AI just tying together a bunch of thousands of disparate papers and pulling at the threads — sorry for bringing back the metaphor — but the threads of the different papers, and then just basically articulating the proofs that were already there in the different papers and you're making minor connections? Or is it actually developing new mathematical ideas? Have you seen glimmers of whatever that genius AlphaZero move in the game of Go?

对，第 37 手。

Yeah, move 37.

第 37 手。

Move 37.

我还没读完他们全部 700 篇论文。我看过的那些——有些人对此很不屑，说，“哦，那都在凸包里面。你做人已知的那些动作，只是做得非常好，做了很好的树搜索。”我不会把它叫作凸包。我觉得 Tim Gowers 有个很好的说法，把它称作由所有这些生成的一个半群。是的，它还是把已知动作组合起来的概念，但已知动作的空间、它能达成的东西，比那种带贬义的凸包要大得多得多。所以就准黎曼猜想而言，它走的那些步都是人类已知的东西。而其中有很长的一连串，当你看着它时会感到震惊，说，“天哪，太美了。你怎么想到——当然，我是在拟人化——你怎么想到在这一步之后走那一步，再之后走那一步？”是的，就是这样。

So I haven't read all 700 of their papers. The ones that I have looked at — some people speak about this dismissively, like, "Oh, it was in the convex hull. You do the moves that we know as humans and you just do them really well and do a really good tree search." I wouldn't call it the convex hull. I think Tim Gowers had a really good way of talking about it instead as the semigroup generated by all of these things. Yes, it's the same notion of putting together known moves, but the space of known moves, what it can accomplish, is much, much bigger than this kind of derogatory convex hull. So in terms of the quasi-RH, the moves that it took are all things that were known to human beings. And there's a very long cascade of them that is kind of shocking when you look at it and say, "Oh my god, that's so beautiful. How did you think to — I'm anthropomorphizing it, of course — how did you think to do this move after that move and after that move?" And yeah, that does it.

你能向普通大众解释一下直觉吗？半群和凸包之间的界限，那个带有贬义的凸包，当然还有准黎曼猜想，那是重大证明之一？

Could you articulate to the more general public the intuition, the line between semi-group and convex hull, the derogatory convex hull, and the quasi RH, of course, is the quasi Riemann hypothesis that one of the big proofs?

好，我试着这样解释。你想象一个气球。气球内部就是它的凸包。所以我们有所有这些结果，所有数学家想出来的这些想法。然后里面的任何东西都是它们的某种线性组合。但它感觉是紧致的。就像有一个封闭的区域，而在那个区域内，AI 并没有真正取得新进展，因为它没有把气球往外推。它还没有给我们一个新想法。

Yeah, let me try to do it like this. You have Think of like a balloon. So, the inside of the balloon is its convex hull. And so, we have like all of these results that all these ideas that mathematicians have thought up. And then anything inside of that is some linear combination of it. But it feels compact. It's like there's this kind of closed enclosed area, and within that area, like AI hasn't really made new progress cuz it hasn't moved the balloon out. It hasn't given us a new idea yet.

说它带有贬义，是因为那里的直觉是，我们本可以做到，数学家本可以做到，因为它已经相当明显。它已经在已经思考和发展的空间之内了。

And it's derogatory because the intuition there is we could have done it mathematician could have done it because it's already quite obvious. It's already within the the space of things that have already been thought about and developed.

是的，它读过了每一篇论文。这正是 Ankur 刚才说的。它读过了每一篇论文。它把所有这些都内化了。这比任何人类大脑能容纳的还要多，但人类有时间去阅读、学习和内化。所以，在广度上，是的，它击败了任何人类。然后在深度上，它真的产生了一个真正的新想法吗？你可以争论，至少在准黎曼猜想的情况下，我不会说它是一个真正的新想法。它是许多非常著名事物的美丽组合。有很多人正在懊悔，他们 20 年前就有过那个方向的想法。两年前，在三次 theta 函数和三次高斯和上有一个重大突破，最终成为了 AI 通向解决方案的路径。但为了与半群对比，我把半群想象成像瓷砖。所以，你只是在棋盘上玩瓷砖，你被允许做的是，每当你有一块瓷砖，你可以把另一块瓷砖放在左边、右边、前面或后面。然后你就可以开始铺东西了。这就像半群作用。问题是，你能达到多远？你可以走，我不知道，你可以用现有想法走得非常非常远，可能证明比我们人类能够探索的多得多。

Yeah, so it has read every paper. It's exactly what Ankur was just saying. It has read every paper. It has internalized all of that. That's more than any human brain not even could hold, but has time to read and learn and internalize. So, in terms of breadth, yes, it beats any human being. And then in terms of depth, you know, has it really generated a genuinely new idea? And you can argue that at least in the case of quasi RH, I would not say it's a genuinely new idea. It's a It's a beautiful combination of a lot of things that were very well known. There are a lot of people who are kicking themselves that had some idea of in that direction 20 years ago. And and two years ago there was a big breakthrough on cubic theta functions and cubic Gauss sums and you know, that ended up being that the path that the AI took to to the solution. But so to contrast that with a semigroup, I think of a semigroup as like tiles. So you you know, you just playing tiles on the board and what you're allowed to do is whenever you have a tile you can put another tile to the left, to the right, to the front or to the back. And and then you can just start tiling things. And that's like a semigroup action. You and and the question is well, how far can you reach with that? You can go I don't know, you know, you can go really really far out with with existing ideas might prove a whole lot more than we as humans have been able to explore.

所以你的直觉更倾向于半群方向，意思是至少有一丝扩张的迹象，而不是凸包内的线性组合。

So your intuition is more in the semigroup direction, meaning like there is at least glimmers of expansion versus linear combinations inside the convex hull.

你知道，我们如何选择解释它正在做的事情，我认为取决于我们，但在整个数学史上，有一些看似非常困难和深刻的问题，然后有人想出了一个三页的解决方案，非常初等，本可以向本科生解释，并且本可以在问题首次提出时的 20 年前被发现，如果它在 20 年前被发现，它就会是某本教科书中的练习。因为那个解决方案没有被发现，而真正聪明的人研究了很长时间并取得了渐进进展，使这个问题非常出名，所以当它真正被解决时，它就成了一个大事件。这在数学中时不时发生。一年前我写的时候说过，所以今年夏天有一个国际数学家大会，我在写我的会议录，他们要求提前 10 个月。所以我说，你知道，我现在要写关于数学和 AI 的什么，10 个月后仍然相关？我认为那里的内容有些常青。我坚持我所写的。但是，我在那里写的一件事正是，我们将发现，更多我们认为当前技术无法解决的问题，实际上可以通过当前技术解决，如果你能以正确的方式组合起来。这就是 AI 擅长的地方。再次，有人可以解释为，哦，所以它在里面，这些解决方案在凸包内。你想这么称呼，没问题。我认为那并非无效。

You know, how we choose to interpret what it's doing I think is is up to us, but throughout the history of mathematics there have been very seemingly difficult and deep problems that then someone comes up with three-page solution that is very elementary and could have been explained to an undergrad and could have been discovered 20 years before when the problem was first posed and had it been discovered 20 years before, it would be an exercise in a textbook somewhere. And because it that solution wasn't discovered and really bright people worked on it for a long time and made incremental progress and made the problem really famous, then by the time it was actually solved, it was a really big deal. That happens from time to time in mathematics. And what I said a year ago when I wrote so there was this International Congress of Mathematicians this summer and I was writing my proceedings that they wanted 10 months in advance. So I said, you know, what am I going to write about math and AI now that will still be relevant 10 months from now? And I I think what's there is is somewhat evergreen. I stand by by what I wrote. But, one of the things that I wrote there was exactly that we're going to discover that a whole lot more problems that we thought were unreachable by current technology actually are reachable by current technology if you can put it together in just the right way. And that's what AI is amazing at. Again, someone could interpret that as, oh, you so it's inside, you know, these solutions were inside the convex hull. You want to call it that, that's fine. I think that's, you know, it's it's not invalid.

我喜欢所有这些比喻，还有气球。气球的重要之处在于内部和外部之间的边界。让它带有贬义的是，你似乎没有改变边界。但是，我确实，这些问题已经改变了边界。也许更具体的事情是，在 OpenAI 那批问题之后，人们将要研究的问题类型永远不同了。那里至少有几个结果，是改变领域的结果，会改变你给年轻研究人员什么问题类型，你想探索什么领域，因为你对景观的构想已经大不相同了。

So, I I like all these metaphors, you know, also the balloon. So, the important thing about the balloon is the boundary between what's inside and outside. And what makes it derogatory is the fact that you know, you haven't seemingly changed the boundary. But, I I, you know, definitely these problems have changed the boundary. And, you know, maybe a more concrete thing is just that the types of things that people are going to work on after that drop of OpenAI problems is forever different. You know, there were at least a couple of results there that are like, you know, field-altering results that will change what types of problems you give to young researchers, what areas you want to explore because your conception of the landscape is just that much different.

所以，不仅仅是因为它扩展了，而是因为扩展本身改变了有趣问题的方向。

So, not just the because it expanded, but because the very expansion changed the direction of the interesting questions.

没错。所以，事实上，其中一个问题，也许最容易解释它对人们产生了什么样的影响，实际上不是来自 OpenAI 的那批问题。而是一个 Entropic 的结果，由我的同事 Virgi Vasilievska Williams 和 Josh Alman 撰写。这是关于三数之和问题的。我认为它对所谓的细粒度复杂性理论有重大影响，因为有一个很大的惊喜，你在编码面试中给人们的一个非常基本的问题，实际上可以渐近地比任何人知道的解决方案快得多地解决。这与社区的赌注相反。事实上，有一个完整的理论建立在这样的想法上：如果这个问题真的那么难，那么所有其他问题都继承了它的难度。所以现在，由于这个问题比我们想象的更容易，有一个稍微更好的算法，是的，这给了你整个领域很多新算法。但它也改变了领域的特征，因为那里的目标是把所有这些问题放在一起，说我们应该一起研究它们。但现在我们有点回到了这样的状态：也许研究它们的正确方式是分开研究。所以我认为这些例子表明，无论我们得到多少深刻的新的可推广的想法，即使只是知道定理的真相，也会改变人们在该领域组织研究的方式。而且，其中一些问题已经没有回头路了。

That's right. So, in fact, you know, one of the problems which probably, you know, maybe it's the easiest to explain what kind of effect it had on people is not actually from the OpenAI drop. It was an Entropic result that was written up by my colleague Virgi Vasilievska Williams and then Josh Alman. So, this was about the threesome problem. And I'd say that there it has a major effect on what's called fine-grained complexity theory because there was a big surprise that, you know, a very basic problem that you give people in like coding interviews can actually be solved asymptotically much faster than the solution anyone knew. And that was counter to what the community was betting. In fact, there was a whole theory built up around the idea that if this problem really is that hard, then all these other problems inherit its hardness. So now as a result of that problem being easier than we thought, you know, having a bit better of an algorithm, you know, yes, that gives you a lot new algorithms across the entire landscape. But it also changes the character of the field because the goal there was to put all those problems together and say that we should be studying them all together. But now we're sort of back in the state of affairs where maybe the right way to study them is separately after all. So I think these are examples where, you know, irrespective of how much, you know, deep new generalizable ideas we get out there, even just knowing the, you know, truth of the theorem is going to change the way that people organize research in that area. And, you know, there's no putting the genie back in the bottle on some of these problems.

我的意思是，想象一下 700 个问题中有一个是多项式时间分解。那绝对会改变一切，所有人都会知道。

I mean, imagine one of the 700 was uh factoring in polynomial time. That would absolutely change Everybody would know.

是的。

Yep.

比如，你不能再在亚马逊上用信用卡了。所有电子商务都消失了。所有安全通信都消失了。你能解释一下多项式时间内的因数分解吗？

Like, you can't use your credit card on Amazon anymore. All of e-commerce is gone. All of secure communications is gone. Can you explain factoring in polynomial time?

是的，所以如果你给我两个非常大的数，每个一百万位，我可以瞬间把它们乘起来。如果你让我反过来，比如你取两个大质数，把它们相乘，给我乘积，然后让我找出是哪两个数，就是因数分解。我们不是让孩子在学校学因数分解吗？那是假的。我们不知道如何因数分解。而我们不知道如何因数分解，正是为什么我们有能力以各种安全的方式相互通信。所以所有现代，基本上现代世界都依赖于我们没有一种算法来取出一个大数的质因数。

Yeah, so if you give me two very large numbers, a million digits each, I can multiply them together and do that in a split second. If you ask me to go backwards, say you took two large prime numbers, you multiply them together, you give me the product, and you ask me to figure out which two numbers, just factor. Don't we ask our kids to learn how to factor in school? It's a lie. We do not know how to factor. And the fact that we don't know how to factor is why we have the capacity, the ability to communicate with one another in all kinds of secure ways. So all of modern, basically the modern world depends on the fact that we don't have an algorithm for taking a large number and pulling its prime factors out of it.

所以，很多电子商务确实如此，因为这是公钥加密中许多方案的基础。但众所周知，有用于因数分解的量子算法。所以，如今人们非常重视寻找后量子密码学的基础。

So, a lot of e-commerce does because that's the basis for a lot of the schemes in public key encryption. But, you know, famously there are quantum algorithms for factoring. So, there's a lot of emphasis these days on finding post-quantum bases for cryptography.

基于格的方法。是的，我们已经担心当量子计算机足够大时会发生什么，但我们还有一点时间，在量子计算机变得足够大、使这成为实际问题之前。但没有理论上的原因，我的意思是，好吧，因数分解可能不是 NP 难的。但假设证明 P 不等于 NP。也许那会很了不起，显然，但那不会，如果它证明 P 等于 NP，那将完全

Lattice-based methods. Yeah, we're already concerned about what will happen when we have quantum computers large enough to, but we still have a little bit of time before quantum computers get large enough to make this an actual problem. But, there's no theoretical reason, I mean, okay, factoring's probably not NP-hard. But let's say prove that P doesn't equal NP. Maybe that would be amazing, obviously, but that doesn't, if it proves that P equals NP, that would be completely

那会非常有趣。

That would be very interesting.

我会在复杂性理论的背景下更广泛地问，但想必 OpenAI 或任何其他公司不会轻率地发布那个结果。那种事情应该经过

I'll ask about more broadly in the context of complexity theory, but it presumably OpenAI or any other company would not drop that result carelessly. That's the kind of thing that should go through

是的，政府和

Yeah, the government and

有时某些数学结果相当明显地会对世界产生巨大影响。其他结果可能需要几个月来消化，但那个结果会，是的，会影响整个国家的经济系统，所以。

Sometimes certain mathematical results quite obviously lead to dramatic impact on the world. Other results might take maybe months to digest, but that one would just, yeah, would affect the economic systems of entire nations, so.

NSA 雇佣了多少数学家？

How many mathematicians does the NSA employ?

数千名。

Thousands.

数千名，就像这样。我一直做这个思想实验，你知道，他们可能有什么他们不会告诉我们的东西。这是 10 年前，有证据表明，数学家团队如果以正确的方式分解问题，可以解决比任何单个数学家难得多的多的问题。所以，我不知道。你介意我重新讨论一下气球吗？因为我实际上更喜欢半群类比而不是凸包，但我不认为它应该被解释为贬义，即使它只是说这些机器是在内插人类知识而不是外推。因为即使存在这个思想和事物的凸包，可以从人类发明的碎片中拼凑起来，我们也没有足够快地做到这一点，或者你知道，有很多不透明性，这个气球不是透明的。所以，即使你可以使这个凸包更加透明，就像拨开迷雾，给数学家雷达，让他们看到你知道的地平线之外的东西，那也相当不可思议。我不认为我们应该将其解释为贬义。我确实认为它更具外推性。所以。

Thousands, like something like this. I've always done this thought experiment, you know, what might they have that they just wouldn't tell us about. And this was 10 years ago, you know, that some evidence showed that collaborative teams of mathematicians could solve much, much harder problems than any one individual mathematician if they like broke it down in the right way. So, I don't know. Do you mind if I just revisit the balloon for a second cuz I actually like the semi-group analogy better than the convex hole, but I don't think it's as it should be interpreted as derogatory as it is even if it was just that these machines are interpolating human knowledge rather than extrapolating. Because even if there is this convex hole of ideas and things that could be pieced together from bits that humans have invented, we're not doing that fast enough or you know, there's a lot of opaqueness and it's this balloon is not see-through. And so, even if you can make this convex hole much more transparent, kind of like lift the fog, give mathematicians radar so they can see what's you know, over the horizon, that's pretty incredible. And I don't think we should be interpreting that as derogatory at all. I do think it is more extrapolatory than that. So.

是的，我们应该说可能是凸包的大小。我们实际上不知道它有多大。

Yeah, we should say probably the size of the convex hole. We don't know actually how big it is.

嗯，这就是部分重点。我们不知道它有多大。我们不知道什么与什么相邻。就像我们知道的东西有很多迷雾，而揭开它很令人兴奋。

Well, that's part of the point. We don't know how big it is. We don't know what's next to each other. Like there's so much fog of what we know like and and that's exciting to lift that.

但人类确实会做出新的定义，你知道，引入新概念，这从根本上感觉不同于组合现有想法并重新包装它们。当人类这样做时，他们不时这样做，

But human beings do make new definitions, you know, bring new concepts in and that fundamentally feels different from combining existing ideas and repackaging them. That when human beings do that and they do that from time to time,

是的。

Yep.

真的感觉像是，好吧，有一个新的边界点，现在你能证明的凸包是什么。所以，就像你可以拿起气球，有办法拉扯它。

it really feels like okay, there's a new boundary point and now what's the convex hole of that of what you could prove. So, it's like you can take the balloon and there are ways to pull on it.

我们知道它何时膨胀。就像我们 somehow 有一种直觉它何时变大。就像你能识别出来。

And we know when it expands. Like somehow we have an intuition for when it's getting bigger. Like you can recognize that.

在许多方面，研究者的原型也将因为这项新技术而改变。你知道，我总是惊讶于，当你回顾历史时，非常不同的数学领域实际上源于相似的想法。许多这些联系在开始时推动了发展，但随着时间的推移，它们变得更加专业化。我认为 certainly 我们训练研究生的方式，我们需要确保他们不会以同样的方式过度专业化。你知道，我不知道未来的招聘会是什么样子。我不知道未来的期刊会是什么样子。但我认为，那些广泛思考更大范围数学的人将处于最有利的位置来从这些技术中受益。这是我的猜测。

In many ways, the archetype of what the researcher looks like is going to change because of this new technology, too. You know, I'm always amazed that when you look back historically, how much very different fields of math really spawned from similar ideas. And a lot of those connections drove the development in the beginning, but over time they become a bit more specialized. And I think certainly the way that we train graduate students, you know, we'll need to make sure that they're not just over specialized in the same way. You know, I don't know what the future of hiring will look like. I don't know what the future of journals will look like. But, I think people who think broadly about, you know, a bigger swath of mathematics will be in the best position to benefit from these technologies. That would be my guess.

好的，让我们问一个听起来几乎愚蠢的问题，但几十年来研究的很大一部分是论文。如果你要猜测在未来几年，这会如何改变人类贡献想法的方式，也许合作，一个人类团队如何将想法贡献到公共空间？

Okay, let's go almost silly-sounding question, but so a large part for many decades of research has been the paper. If you were to guess in the years to come, how does that change how a human contributes their idea, maybe collaborate, how a team of humans contribute an idea into the public space?

所以，我的意思是，我不知道纯数学的情况，但在工程和一些应用科学中，我们已经对学术贡献是什么有了相当快速变化的看法。所以，如果你在更大的基础设施中编写可重用的代码，支持更大的项目。想想 LIGO 或一些大型的、数十亿美元的天文项目或癌症研究项目，这些项目已经需要数百人和巨大的资源。它不一定是一个 15 页的 PDF。你知道，有很多不同的部分，代码、数据、可重用的片段可以融入更大的框架。在我看来，这显然与 AI 更相关，拥有一个小巧可用的部件或小工具或 API 或代码库或数据集，可以被社区或 AI 利用，越来越有价值，甚至比仅仅一个结果或一篇论文更有价值。我不知道这在纯数学中会是什么样子。

So, I mean I don't know about in pure math, but in engineering and some of the applied sciences, we already have a pretty rapidly shifting view of what an academic contribution is. So, if you write code in part of a larger infrastructure that is reusable, that supports larger projects. So, think of LIGO or some of the big, billion-dollar astronomy projects or cancer research projects that require hundreds of people and huge resources already. It's not necessarily one 15-page PDF. You know, there's lots of different pieces, code, data, reusable bits that can be fit into a larger framework. And in my mind, that is kind of obviously even more relevant with AI is having a little workable piece or widget or API or code base or data set that can be leveraged by the community or by AI is increasingly valuable, even more than just like one result or one paper. And I don't know how that looks in pure math.

你能谈谈你对一切如何变化的看法吗？甚至只是观察 arXiv 的疯狂演变，这在某些方面类似于 GitHub 上发生的事情。例如，开源运动中的很多人真的不知道该怎么办。有些人预测 GitHub 会消亡。

Can you comment on your view how everything's changing? Even just observing the insane evolution of even arXiv, which is similar in some ways to what's happening on GitHub, for example. A lot of people in the open source movement are really struggling to know what to do. Some are predicting the death of GitHub.

为什么？只是因为添加到 GitHub 的代码太多，无法维持增长吗？

Why? Just cuz there's so much code being added to GitHub that it can't sustain the growth?

所以，开源运动成功的因素之一是每个仓库都有一两个真正管理的人，这些存在，他们在心理上和许多层面上都被所有的拉取请求淹没了。所以，变得无法管理，他们筋疲力尽，有一种整个的感觉，

So, one of the things that makes the open source movement successful is there's a kind of one or a few folks that really manage each repository, the beings, and they are, um, psychologically and on many levels are becoming overwhelmed by the all the pull requests. So, it becomes impossible to manage and they get exhausted and there's this whole feeling,

是的。

Yeah.

开源运动，也许类似于学术环境，需要人类有动力。如果你被类似陶哲轩提到的想法所累，你就会退出。所以，这是一种你想避免的失败模式。

the open source movement, maybe similar to the academic environment, requires the humans to be motivated. And if you get worn out by the just similar the ideas that Terry Tao mentioned, that you just quit. And so, that's a failure mode that you want to kind of avoid.

所以，数学期刊在 AI 出现之前就已经因为数量问题而承受着非常非常严重的压力。与 AI 无关，只是有更多的人产生更多的知识，发送给期刊，期刊必须进行审稿，审稿人筋疲力尽，被要求非常努力地理解这些定理并检查它们。我认为，唯一比那个系统更糟糕的事情就是没有那个系统。因为我们的数学认识论，我们达到真理的方式，是故意缓慢、仔细和精确的。你可以回顾佩雷尔曼对庞加莱猜想的解决方案。他把这些论文放在 arXiv 上。然后世界各地的人都被震惊了。所以他们邀请他来做讲座。他来到石溪，来到纽约大学，来到普林斯顿。他走遍全国和世界。他做 3 小时长的即兴解释，也许一个小时是他解释他在做什么，接下来的两个小时是人们说：“在第 247 页，引理 412，我不明白你是怎么做的。”然后他走到黑板前，给他们做一个半小时的讲座，讲解他是怎么做的。所以整个事情活在他的头脑中。他已经尽力在纸上沟通。然后由我们社区来非常仔细、非常详细和精确地回顾，并同意：“是的，这是庞加莱猜想的真正证明。”这花了数年时间和大量的人力。所以提出解决方案只是过程的第一步。这个过程必须被交付，并且必须由社区驱动。这就是期刊系统试图实现的目标。

So, journals were already in math, uh, under very, very severe strain from the number of things before AI. Having nothing to do with AI, just there was, you know, more people producing more knowledge, sending it to journals, and journals had to referee it, and referees are exhausted and being asked to, uh, work really hard to understand these theorems and check them. Uh, the only thing worse than that system, I think, is not having that system. Because the way that like our mathematical epistemology, the way that we arrive at truth, is deliberately slow and careful and precise. You go back to Perelman's solution of the Poincaré conjecture. He puts these papers on arXiv. And then people around the world are, you know, shocked by this. And so they invite him to come give lectures. He gives He comes to Stony Brook. He comes to NYU. He comes to Princeton. He's going all around the country and the world. And he's giving 3-hour-long impromptu explanations of of, you know, maybe an hour is is him explaining what he's doing and the next 2 hours are people saying, "On page 247, lemma 412, I don't understand how you do this." And then he goes to the board and gives them a half-hour lecture on how he does that. So the entire thing lives in his mind. He's done the best he can to communicate it on paper. And then it's up to us, the community, to go back through it very carefully, very in a detailed and precise way, and agree, "Yes, this is a real proof of the Poincaré conjecture." And that took years to do and a whole lot of human effort. So coming up with the solution is only the first step in the process. And that process has to be delivered and it has to be community driven. And so that's what the journal system is trying to accomplish.

所以你提出了一个非常有趣的观点，它突显了纯数学家的传统形象：一个孤独的天才，解决一个独特的问题，然后向世界解释。在许多其他领域，我们已经超越了个人无法推进的阶段，比如电路设计、飞机设计、蛋白质折叠或天文学。我们必须在一个数十亿美元的合作项目中一起工作。而且，我是一名工程师，所以我认为领导力显然是社区在面临组织危机和从个体性转向更集体模式时所需要的。这很有趣，因为我认识很多数学家，从组织上来说，这是一个非常独立的群体，他们不喜欢被告诉该做什么，不喜欢被告诉该研究什么，他们天生非常独立。我们喜欢独处和独自工作。这就是为什么我们花那么多时间这样做。

So you make a really interesting point and it kind of highlights this traditional view of a pure mathematician as this like lone genius who you know, solves this singular problem and then goes explaining it to the world. In many other fields, we have moved past the point where we can't individually advance, you know, circuit design or aircraft design or protein folding or any you know, astronomy. And we have to work together in these billion-dollar collaborative projects. And again, I'm an engineer, so I'm thinking leadership is obviously the thing that a community needs when it's having this kind of crisis of organization and individuality moving towards a more collective model. Um, and that's interesting because I know so many mathematicians and like by kind of organization, this is a very independent group of people who don't like being told what to do, who don't like being told what to study, who are very independent by their very nature. Um, you know, we like solitude and working alone. That's why we, you know, spend so much time doing it.

我认为这是一个非常好的观点。我的意思是，看到工作风格的差异很有趣，甚至数学和物理之间也是如此，比如大型强子对撞机。所以，让我对这些事情采取一种逆向观点，那就是，你不应该浪费一场好的危机。我同意期刊超负荷，会议超负荷。总是让我震惊的是，我们做了那么多工作来审查这些东西，处理所有这些事情。现在我们面临的任务是有人必须审查所有 AI 生成的论文。其中一些很重要，但真的很难让你的社区参与进来。我真的，我不知道会发生什么，但我真的赞成寻找新的模式，不是取代旧模式，而是缓解一些已经出现的压力。所以，我很想看到这样的模式：也许我们采取更合作的方式来验证一些 AI 证明，对吧？所以，OpenAI 发布的一些结果，比如快速傅里叶变换的改进。有推文流中人们改进这个概念。我真的不知道该怎么理解，我甚至不在 Twitter 上。在我看来，我们需要找出其他工作模式，用于组织所有改进的状态，但也如何组织和驯服这些想法？

I think that's a really good point. I mean, it's interesting just to see the differences in working styles, even between math and physics, you know, with things like the Large Hadron Collider. So, I mean, let me take a contrarian view for these things, which is, um, you know, you should never let a good crisis go to waste. I agree that journals were overloaded, the conferences were overloaded. It always strikes me that we're doing so much work, you know, to review these things, to handle all these things. And now we're faced with a task that someone's got to review all these AI-generated papers. And some of them are important, but, you know, it's really hard to get your community on board with doing that. I would really, you know, I don't know what's going to happen, but I would really be in favor for finding new models, not to replace the old ones, but to relieve some of the pressures that have come about. So, you know, I'd love to see models where, you know, maybe we take a more collaborative effort to uh verify some of these AI proofs, right? So, some of the OpenAI results that have dropped, you know, improvements on fast Fourier transform. You know, there's been tweet streams with people improving the concept. I don't really know what to make of that, you know, I'm not even on Twitter. Um, you know, and it seems to me that we need to figure out other modes of working, you know, for organizing, what are the state of all the improvements, but also how do we organize and tame these ideas?

嗯。

Mhm.

所以，有点引人注目的是，我们对出版物的看法是，你努力写草稿，发布在 arXiv 上，那就是你的草稿。你可能会在某个时候更新并创建一个新版本或第三个版本，但如果草稿属于你，社区并不真正负责添加注释。嘿，我认为你可以用不同的方式做这个论证。嘿，我认为这样简化了。但是，也许我们需要其他模式来做某些事情，比如允许众包来让我们的理解进化。我完全赞成。另一件事是我喜欢做研究演讲。你知道，做研究演讲我最喜欢的部分之一是你可以把论文中所有混乱的复杂性，那些你只知道必须注意细节的地方，但那并不是真正让你兴奋的。有一些关键想法，有一些特殊情况真正说明了为什么那个想法强大，那才是你真正想向人们解释的部分，因为当有人问我引理 57 时，我甚至不记得了，对吧？也许，这个想法，即论文只是一个单一的东西，设计成写下来并完全形式化。

So, it's kind of striking that, you know, the way that we think about publications as you work hard on your draft, you post it on arXiv, and that's your draft. You know, you might update and create a new version at some point or a third version, but if the draft belongs to you. It's not really up to the community to add notes. Hey, I think you can do this argument differently. Hey, I think this is simplified this way. But, maybe there's some kinds of things that we need other models for, you know, allowing uh crowd-sourcing for our understanding to evolve. I'd be all for that. You know, the other thing is I I love giving research talks. Um you know, one of my favorite parts about giving research talks is you can take all of the messy complexities of a paper, the things where you just knew you had to dot the eyes and cross the tees, but that wasn't really what excited you. There was some key idea, and there was some special case that really illustrated why that idea was powerful, and that's the part you really want to explain to people because when I someone asked me about lemma 57, I don't even remember it, right? And maybe, you know, this idea that um uh a paper is just this monolithic thing designed to be, you know, written down and completely formalized.

现在，我们可以在一定程度上信任 Lean 形式化对于某些技术构建块的处理，这可以让我们重新设计写作方式，将更多重点放在这些思想的传播上。

Well, now we can trust Lean formalizations to some extent for some of the technical building blocks, and that can allow us to rework the way that we do the writing in the first place, that we can put more of the emphasis on the work on the promulgation of those ideas.

MathLib 模式正是你所描述的，不是论文，而是 PR，一个拉取请求。维护者和审稿人会审查拉取请求，他们会提出各种建议。我的意思是，审稿人会提出建议，你可以改进这个那个，但这不是期刊中的公开过程。在 GitHub 开放工作区，你可以确切看到谁贡献了什么想法简化了论证，或者你在重用这个，你在重新发明库中已存在的想法。你应该直接引用那个并重构这部分，等等。所以，MathLib 正式工作首次允许——数学的问题之一是我们没有为定理和技术引理编写完美的规范。如果你随机打开一本数学书的任意一页读一个引理，有太多上下文

The MathLib model is exactly what you were describing, where instead of a paper, you have a PR, you have a pull request. The maintainers and the reviewers will go through the pull request, they'll make suggestions here and there. I mean, the referees make suggestions, you can improve this and that, but it's not a public process in a journal. In GitHub open workplace, you can see exactly who contributed what idea that simplified an argument or you're reusing this, you're reinventing this idea that already exists in this part of the library. You should just quote that and refactor this part, whatever. So, and MathLib working formally allows one for the first time—like one of the problems with mathematics is that we don't have a perfectly well-written specification to our theorems, our technical lemmas. If you open up a random page in a random math book and read a lemma, there's so much context

嗯。

Mhm.

作者——因为填写每个上下文细节太烦人了。所以我们假设你已经读了前 60 页。现在我可以直接告诉你那个引理的下一步。但当你尝试——这意味着你不能像建造火箭时那样直接使用现成的物理部件。

that the author—because it's so annoying to write something that fills in every bit of context. So we just assume you've read the previous 60 pages. Now I can just tell you what the next step is in that lemma. But then when you try to—that means you can't just take it off the shelf the way you can a physical part when you're building a rocket.

嗯。

Mhm.

所以如果未来所有数学都正式完成，那么规范就内置在对象中。你知道，你不会以错误的方式使用它，因为你不能以错误的方式使用它，因为 Lean 不会允许你。这就是众包版的数学。这就是 MathLib 的梦想

And so if all of mathematics is done in the future formally, then the specifications are built into the object. You know, you're not using it in the wrong way because you can't use it in the wrong way because Lean won't let you. And that is the sort of crowd-sourced version of mathematics. And that was the dream of MathLib

嗯。

Mhm.

当它 10 年前启动时。就是像科学一样运行数学。

when it was started 10 years ago. Was to run mathematics more like a science.

顺便问一下，你能提一下 MathLib 和 Lean 是什么吗？因为我们为听众提了几次。

As a quick aside, can you mention what MathLib is and Lean just because we mentioned it a few times for folks?

当然。所以数学理想上是一个形式化过程，从公理化系统开始，然后从那里进行逻辑推理规则。所以如果人们想到——我不知道，人们还在高中学习欧几里得几何吗？那应该是孩子们第一次看到证明和公理结构的模型。

Sure. So mathematics is in the ideal a formal process starting with an axiomatized system and then logical rules of inference from there. So if people think of—I don't know, do people still learn Euclid's geometry in high school or something? That's supposed to be the kind of model for kids to start seeing proofs for the first time and seeing an axiomatic structure.

嗯。

Mhm.

你可以把这个想法写进软件，说这里是数学的实际公理，这里是逻辑推理规则，而 Lean——这类软件有很长的历史，叫做交互式定理证明器，你在证明中逐行编写，计算机说是的，这一行确实从上一行的目标状态通过逻辑推理得出。所以，Lean 是允许你检查各种东西的软件。它可以检查你代码的规范是否符合你试图定义的内容。这就是它被发明的原因——为了无 bug 的代码，希望未来我们能有无 bug 的代码，因为 AI 系统非常擅长发现人类编写的代码中的 bug。所以，对于那个应用来说，它变得越来越重要。但另一个应用是检查数学结果。并允许你以非常正式的方式编写数学结果。所以，现在 Lean 本身只是检查逻辑步骤的硬件。你不想每个数学项目都从定义自然数开始，使用零和后继——你知道，自然数的皮亚诺公理。所以，有一个大型的通用库，维护得非常好，组织得非常好，等等，是很好的。这就是 MathLib。MathLib 是 Lean 生态系统中用于形式化数学的 Lean 库。再次，Lean 只是一个交互式定理证明器，还有很多其他的。它是到目前为止在数学战争中获胜的那个，如果有数学战争的话——这是两个人的竞争之类的。

And you can take that idea and write it in software and say here are the actual axioms of mathematics, here are the rules of logical inference, and Lean—there's a long history of this kind of software, it's called interactive theorem provers, where you're writing line by line in a proof, and the computer says yes, this line does follow by logical inference from the goal state on the previous line. So, Lean is software that allows you to check all kinds of things. It can check the specifications of your code align with what you were trying to define. That's what it was invented for—was bug-free code, and hopefully in the future we will have bug-free code because AI systems are really good at finding bugs in code written by humans. So, it's becoming ever more important for that application. But another application is to check mathematical results. And allow you to write a mathematical result in a very formal way. And so, now Lean itself is just the hardware that checks the logical steps. You do not want to start every math project by defining what the natural numbers are, using zero and the successor—you know, the Peano axioms of the natural numbers. So, it's nice to have a large general library that's extremely well-maintained, extremely well-organized, and so on. And that's what MathLib is. MathLib is the Lean library for formalizing mathematics in the Lean ecosystem. And again, Lean is just one interactive theorem prover, there are lots of others. It's the one that so far has been winning the math wars, if there are any math wars—it's a competition of two people or something.

而且它是部分用于展示 OpenAI 发布的所有不同证明的正确性的技术。

And it's the technology used in part to show that the OpenAI release of all the different proofs, that the proofs are correct.

没错。现在已经变得标准——因为自动形式化已经发展到 AI 系统不仅能用自然语言证明定理，你不能信任它们的自然语言证明，它们是过程。所以，它们说东西并不意味着它们头脑中一定有数学的模型，不管那是什么意思，就像我们一样。但没关系，因为它们会写 Lean 代码，建立在 MathLib 之上，建立在素数定理加项目或其他之上，并检查它确实解决了这个问题，直到公理。

That's right. It has become now standard to—because auto-formalization has gotten to the point where AI systems can not only prove a theorem in natural language, you can't trust their natural language proofs, they're processes. And so, the fact that they're saying things doesn't mean that they necessarily have a model of mathematics in their heads, whatever that means, the same way that we do. But that's okay, because they'll write a Lean code that builds on top of MathLib, builds on top of the prime number theorem plus project or whatever, and checks that it has indeed solved this problem down to the axioms.

所以，这里我要岔开一下，这个思想实验现在可能有点哲学，但如果你有一个黑箱，给你你无法理解的证明呢？假设更有趣的情况不是因为太大，而是因为想法对任何该领域的世界专家数学家来说都说不通。你怎么办？

So, this is where I'll just take the tangent, and the thought experiment may be a bit philosophical at this moment, but what if you have that black box that gives you proofs that you can't understand? Let's say the more interesting case where not because it's so big, but because the ideas just don't make sense to you to any of the world expert mathematicians in that field. What do you do with that?

所以，几乎根据定义，唯一的方式，因为逻辑推理规则是写下来的，只有少数几个，有一小把。公理，有一小把。所以，你拥有不可理解数学的唯一方式是规模。

So, the only way, sort of almost by definition, because the logical rules of inference are written down, and there are only a few of them, there's a handful of them. The axioms, there's a handful of them. So, the only way that you have mathematics that's not understandable is scale.

让我加一个反方观点。所以，我同样想知道的是，去形式化数学会是什么样子？你知道，形式化确实极大地帮助了我们确信结果是正确的。但是，我们自己理解结果的许多方式实际上是在某种非形式化的层面上。你知道，什么是有趣的特例？你知道，这从何而来？你知道，你使用的这个技术，对吧？而且，你知道，也许那就是那个世界的样子，如果我们有这些黑箱，我们会尝试梳理并弄清楚如何问它们更简单的问题，以发展某种理解系统，了解它们实际上是如何得到答案的。

Let me add a contrarian take. So, one of the things I also wonder is what would unformalizing math look like? You know, certainly formalization helps tremendously that we have confidence that the results are correct. But, much of the way that we understand the results ourselves is really at some unformalized level. You know, what is an interesting special case? What, you know, where does this come from? You know, this technique you're using, right? And, you know, maybe that's what that world would look like if we have these black boxes, we would try to tease apart and figure out how to ask them easier questions to develop some kind of system of understanding of how they actually got to the answer.

是的，所以当我们写论文时，我们混淆了两件不同的事情。一个是证明的工程，另一个是向他人传达。它们几乎相互矛盾。我去年秋天教实分析时有过这种经历，你打开任何实分析教科书，这是我学实分析时的经历。

Yeah, so there are really two different things that we conflate when we write a paper. One is the engineering of the proof, and the other is the communication of that to someone else. And they're almost at odds with one another. I had this experience as I was taught real analysis last fall, and you open up any real analysis textbook, and this was my experience when I took real analysis.

头几节课讲的是实数到底是什么。我们得把它们定义或构造出来，比如有理数柯西序列的柯西等价类，天知道那都是什么意思。一门实分析入门课要花上两周才能讲到那儿。然后我们有了实数轴，现在终于可以开始做微积分了，用正式的 epsilon-delta 定义，诸如此类。这一直让我觉得很奇怪，因为上课之前我就知道实数是什么了。它就是这条数轴，然后我们花两周讲有理数的柯西序列，到了第三周，它又变回数轴了。这到底有什么意义？

The first couple of lectures are about what are the real numbers. We have to define them or construct them as Cauchy equivalence classes of Cauchy sequences of rationals, whatever the hell all that means. It takes like two weeks for a first-semester course in real analysis to get there. And then we have the real number line, and now we can start doing calculus with formal epsilon and delta definitions, yada yada. And it always struck me as bizarre that we do this, because before the class, I knew what the real numbers were. It's this number line, and then we spend these two weeks talking about Cauchy sequences of rationals, and then on week three, they're the number line again. What the hell was the purpose of that?

这是因为过去写东西的方式非常工程化。你得先构造出实数，然后才能开始用它们。但人类早就在用实数了，远早于我们意识到它们需要被形式化地构造出来。所以我在课上做的是直接跳到 epsilon-delta 微积分。我们做了六周的计算，直到某一天，等等，这里有个定理——它在实数里成立，涉及某种收敛性，叫波尔查诺-魏尔斯特拉斯定理。你有一个在固定区间里跳来跳去的实数序列，必然存在某个子序列收敛到一个实数。可这在有理数里不成立。为什么在实数里成立？实数到底是什么？所以，你不碰到一个需要解决的问题，就不会开始问这个问题。

And so, it's because the way things are written, or had been written, was very engineering. You need to construct the real numbers before you start using them. But human beings used the real numbers long before we understood that they needed to be constructed formally. So what I did in my class is I jumped straight into epsilon-delta calculus. We did six weeks of calculations until we got to a point where, wait a second, here's a theorem — it's true in the real numbers, there's some convergence, something called the Bolzano-Weierstrass theorem. You have a sequence of real numbers that are jumping around but in a fixed range; there has to be some subsequence that converges to a real number. Well, that's not true in the rationals. Why is it true in the reals? What are the reals? So you don't start asking that question until you hit a problem that needs to be solved.

所以我教这门课的方式，跟把它当成工程项目来教所需的方式正好相反，因为你不先把实数构造出来、不先知道它们是什么，就不能直接谈实数。如果你是在搭建理论，那没错；但如果你是在讲故事，那你完全可以就从实数开始。你对它是什么有直觉，直到你的直觉崩掉之前，你都不需要知道它究竟是怎么被构造出来的那个分辨率。

So the way I taught the course was sort of backwards from how it needs to be taught if it's an engineering project, because you can't just start talking about the real numbers until you've constructed them, until you know what they are. Well, that's correct if what you're doing is building up the theory, but if you're telling a story, then you absolutely can just start with the real numbers. You have an intuition for what that is, and until your intuition breaks, you don't need the resolution of how you actually construct them.

所以我认为我们未来写论文的方式会非常不同，我们会花更多时间讲这个过程的故事：我在这过程中学到了什么，什么是有意思的——这里、这里和这里你用的是标准做法，但到了这里你卡住了，你往哪儿走？这里有个想法，解释为什么你往左走而不是往右、往上、往下之类。而且有一个 Lean 形式化伴随着这个故事，让你知道工程上确实行得通。因为你不能光讲故事。如果只讲故事，数学不是——它不是音乐，音乐里你可以讲一个漂亮的故事，某处有平行五度、你破了某条规则也没关系，反正巴赫老这么干。

So I think the way that we write papers in the future is going to be very different, where we spend a lot more time just talking about the story of what have I learned in this process, what are the interesting — here you do the standard thing here, here, and here, but then here you're sort of stuck and where do you go? And here's an idea for why you go left and not right and not up and down and whatever. And there's a Lean formalization that accompanies this story that lets you know that the engineering actually works out. Because you can't just tell stories. If you just tell stories, math is not — it's not music, where you can tell a beautiful story and it's okay that you have parallel fifths moving somewhere and you've broken some rule, but Bach does it all the time, so.

我觉得这一点非常重要，正好强调一下 Alex 说的。有一种关于数学的看法，我并不认同，就是把数学看成定理不断上涨的潮水。AI 怎么会不好呢？它在给这股潮水添砖加瓦。但数学真的不只是我们拥有多少条定理，远不止于此。它还关乎与问题搏斗的那个故事。因为当你和一个好问题搏斗时，它会反击你，你会从中有所领悟。它真正关乎的是你沿途想出的那套信念与抽象体系——当你和这个问题搏斗时，你会发现某种更普遍、适用更广的意义。我认为这是一个绝佳的机会，让我们摆脱那种“我们只是一直在公理化，直到抵达下一条定理”的想法。讲出我们究竟是如何走到这些问题上的故事、它们与其他问题有什么相似之处，在这个关键时刻，比以往任何时候都更重要。

So I think that's a very important point, just to emphasize the point that Alex is making. There's this view of mathematics, which I don't believe in, where it's just this rising tide of theorems. How can AI be bad? It's contributing to this rising tide. But really mathematics is not just about the number of theorems we have, far from it. It's also about that story of fighting with problems. Because when you fight with a good problem, it fights back at you and you learn something about it. It's really about the system of beliefs and abstractions you come up with en route — when you're fighting this problem, you discover some more general meaning that applies more broadly. And I think it's a great opportunity to disabuse ourselves of this idea that we're just axiomatizing things until we get to the next theorem. Telling that story of how we actually got to these problems, what the parallels are to other problems, is now, at this pivotal moment, more important than it ever has been.

是的。回到你那个问题，就是，这是不是规模的问题？嗯，是的，就是规模的问题，但正如 Ankur 所说，我们思考数学的方式关乎故事，而可能我们并不理解这件事的故事。你可以逐行跟着一个证明走，却不明白到底在发生什么。你也许可以叫另一个 LLM 给你讲讲这个故事，如果它能给出这样的解释的话。但那正是我们想要的。我们想要故事。我们想要理解。

Yes. To get all the way back to your question about, you know, is it a matter of scale? Well, yes, it's just a matter of scale, but exactly to Ankur's point, the way that we think about mathematics is about stories, and it might be that we don't understand the story of this. You can follow a proof line by line and not understand what's really going on. You might just ask some other LLM to explain to you the story, if it can come up with such an explanation. But that's what we want. We want stories. We want understanding.

当你和问题搏斗时，它会反击。我觉得这句话道出了人类所有伟大追求的一个深刻真理。说到这个话题，我想和你们聊聊史上最伟大的几场搏斗，那些被攻克的数学中最难的问题。但首先，如果方便的话，我们休息一下。

When you fight with the problem, it fights back. I think there's a deep truth to that about all the great pursuits that humanity has taken on. And on that topic, I would like to talk to you about some of the greatest fights of all time with some of the hardest problems in mathematics that have fallen. But first, about the break, if it's okay.

当然。

Sure.

那么 Alex，你是数论学家。也许可以说，你所在领域的核心问题是黎曼猜想。你刚才提到的 Twitter X 上，对那个 quasi-RH、那个准黎曼猜想的证明表现出了极大的兴奋。你能解释一下吗？能解释一下黎曼猜想是什么吗？也能解释一下你兴奋的本质是什么吗？

So Alex, you're a number theorist. Maybe it's fair to say the central problem in your field is the Riemann hypothesis. And you showed a lot of excitement on the aforementioned Twitter X about the quasi-RH, the quasi-Riemann hypothesis proof. Can you explain that? Can you explain what the Riemann hypothesis is? And can you explain the nature of your excitement about it?

当然。黎曼猜想有一分钟版本、五分钟版本，或者三十小时版本。我不知道你想深入到哪一步——

Sure. There's sort of a one-minute version of the Riemann hypothesis, or a five-minute version, or a thirty-hour version. I don't know how far you want to —

我们就从这根线头开始拉吧。

Let's start pulling at that string.

我们知道素数是什么。人们对素数的分布很感兴趣。而黎曼发现的是——这简直疯了——存在一个函数，它以复数为输入，以复数为输出。而这个函数支配着素数的行为。它叫黎曼 zeta 函数。所谓 zeta 函数的零点，指的是那些让输出为复平面上数字零的复数输入，也就是复平面的原点。问题就是，这些复零点在哪里？什么样的复数输入，在你代入这个 zeta 函数后，会得到零这个复数输出？

So we know what the prime numbers are. People are interested in the distribution of the prime numbers. And what Riemann discovered is that there's a function — this is totally crazy — that there's a function that takes complex numbers as inputs and gives complex numbers as outputs. And that function governs the behavior of the prime numbers. It's called the Riemann zeta function. And by the zeros of the zeta function, we mean complex inputs where the output is the number zero in the complex plane, so the origin in the complex plane. And the question is, where are these complex zeros? What complex number inputs give you a complex number output of zero when you apply this zeta function?

这到底为什么会——复数，你在说什么？我们有的是这些在数轴的自然数里、不能被任何其他数整除的离散数字，现在你却要谈从复空间到复空间的函数。这就是那个伟大的突破。伟大的突破在于弄明白了如何对素数做信号处理。你想想，人们能听到你的播客，是因为我的声带在产生声音的压力波。那是自然空间。然后它进入麦克风。

Why in the world should that — complex numbers, what are you talking about? We have these discrete numbers that aren't divisible by anything else sitting in the natural numbers on the number line, and now you want to talk about functions from a complex space to a complex space. That was the great breakthrough. The great breakthrough was to figure out how to do signal processing on the prime numbers. If you think about it, the reason people can hear your podcast is that my voice box is producing pressure waves of sound. That's the natural space. Then that goes into the microphone.

麦克风对它进行分析，做傅里叶分析，把它转换成振幅和频率，就得到一堆数字，存储这些数字或发送这些数字，随便怎样。然后扬声器拿这些数字重建出在人们听来像我的声音。同样的事情也在发生。这就是 zeta 函数对质数所做的。它把质数——你可以把质数想象成空气中振动的分子颗粒——对它做信号处理分析，能够恢复出同样的质数。频率和振幅对应于 zeta 函数的零点。zeta 函数包含了那些信息。而黎曼假设——所有这些的振幅，你可以称之为黎曼波，这些波的叠加恢复了质数的分布。所有这些振幅都应该完全相同。就像演奏质数音乐的交响乐团，每件乐器都以完全相同的振幅发声。这就是黎曼假设所预测的。

The microphone analyzes it, does the Fourier analysis on it, converts it into amplitudes and frequencies, and it's just a bunch of numbers, stores those numbers or sends those numbers, whatever. The speaker then takes those numbers and reconstructs back out what sounds to people like my voice. The same thing is going on. That's what the zeta function does for the prime numbers. It takes the prime numbers—you can think of the prime numbers as particles of air molecules wiggling around—and it does a signal processing analysis on it that can recover the same prime numbers back. And the frequencies and the amplitudes correspond to zeros of the zeta function. The zeta function has that information in it. And the Riemann hypothesis—so the amplitudes of all of these, you can call them Riemann waves, and the superposition of those waves recovers the distribution of the primes. And all of those amplitudes are supposed to be exactly the same. Like the orchestra that plays the music of the prime numbers has every single instrument sounding at exactly the same amplitude. That's what the Riemann hypothesis predicts.

那么，你能否实际上——我们知道质数是什么，但也许进入哲学和浪漫美学观念的领域，为什么它们在数学的核心如此有趣、重要、中心和美丽？而与此相关，黎曼假设之所以令人兴奋，是因为为什么弄清楚质数中是否存在模式以及是什么模式如此困难？

So, can you actually—we know what the prime numbers are, but maybe going into the space of philosophy and the romantic notions of aesthetics, why are they so interesting and important and central and beautiful at the core of mathematics? And connected to why the Riemann hypothesis is exciting is why is it so difficult to figure out if and what the pattern is in the prime numbers?

让我先回答你的第二个问题。为什么某件事困难——我只在知道答案之后才知道如何回答。

Let me start with your second question first. Why something is difficult—I only know how to answer once I know what the answer is.

一旦它变得简单。

Once it's made simple.

困难。也许它并不像我们想的那么困难。我没想到我会在有生之年看到准黎曼假设的证明。所以，我在周二看到它这一事实令人极其震惊。

Difficult. Maybe it's not as difficult as we thought it was. I did not think I would see a proof of the quasi-Riemann hypothesis in my lifetime. So, the fact that I saw it on Tuesday was extraordinarily shocking.

你看到它的那一刻做了什么？比如在 30 秒后——你记得什么——

What did you do the moment you saw it? Like in 30 seconds after—do you remember what—

是的，我知道我当时在哪里。我记得我看到发布的那一刻我在哪里。当时我们在纽约市的国家数学博物馆晚宴上。所以我一手拿着威士忌，不停地刷新——我在和人社交聊天，但我也在不停地刷新，因为我知道它会在东部时间 6 点左右发布。然后发布来了。事实上，我正在和——那里有一位 Quanta 的记者，我们正在聊天。哦，是的，他们要发布这些，你知道，300、400 个——我们不知道有多少，结果是 700 个——问题。我说：“哦，哇，那真的很酷。发布了。”所以，我看着手机，我看到的第二件事就是准黎曼假设。我说：“你说的准黎曼假设是什么意思？你不是说对于什么吧？”我一看，是 zeta 函数，全是狄利克雷 L 函数，所以他们解决了那些等零点。我说，这不可能是这次发布的内容。那只是——那一刻我无法理解我所看到的。

Yeah, I know where I was. I remember where I was the second I saw the drop. So, we were at this National Museum of Mathematics gala in New York City. And so, I had a whiskey in one hand and I kept refreshing—I was talking to people socializing, but I also kept refreshing because I knew it was dropping around 6:00 Eastern time. And then the drop hit. And in fact, I was talking to—there was a Quanta reporter there and we were just chatting. Oh, yeah, they're going to drop these, you know, 300, 400—we didn't know how many, 700 it turns out—problems. And I was like, "Oh, wow, that's really cool. It dropped." So, I'm looking at my phone and the second thing I see is quasi-RH. And I'm like, "What do you mean quasi-RH? You don't mean like for what?" And I look and it's the zeta function and it's all Dirichlet L-functions, so they've solved those equal zeros. And I'm like, that can't be what's in this drop. That just—I could not compute in that moment what I was seeing.

这就是在那短短时间里的进步速度，是的。

That's the speed of progress in that little time, yeah.

只是——它远远超出了我对可能性的模型。而这——这一个月里发生的事情远远超出了我对可能性的模型。费马大定理的形式化，我确信我们至少还需要 3 年才能做到那样的事情。

It's just—it was so far outside my model of what was possible. And this is—it's been a month of things happening way outside of my model of what was possible. The Fermat's Last Theorem formalization, I was sure we were 3 years away at least from being able to do something like that.

所以也许你可以详细说明一下。它是在 Lean 中，它被形式化为——费马大定理在 Lean 中有多大？

So maybe you can elaborate on that. It was in Lean, it was formalized as—how big was the Fermat's Last Theorem in Lean?

1300 万行。

13 million lines.

1300 万行，好的。

13 million lines, okay.

所以就在两天前，我在普林斯顿数学系做了一个学术报告，我详细解释了 Lean 是什么，MathLib 是什么，但也提到当今的 AI 系统——有这样一个 MathLib 库，其中一切都做得完美，恰好达到正确的泛化水平，不应该有任何重用代码——那是在其他地方重新证明定理。一切都是最一般、最抽象的。而 AI 系统能做的就是在上面构建一两层。但它们所做的是临时拼凑。它们还没有被训练——我不是说它们做不到，但它们现在不擅长——我称之为“典范化”的工作，即把有点粗糙的代码重构，准确找出它在库中属于哪里，这个引理的规范陈述版本是什么，然后每个证明都应该只是组合——你应用这里的这个引理，你应用那里的那个引理，把所有这些放在一起，就是你的定理。然后它非常干净，非常美丽，非常优雅。这就是 MathLib 追求的目标。我在报告中说的是，我们今天所处的位置是，你可以让整本教科书形式化，只要它们在 MathLib 的光环内，只要它们与 MathLib 中已经完成的东西相隔一两步。但当你开始——这曾是我的实践，我的经验是——当你开始要求它做与库相隔三四步的事情时，它为这个对象想出的定义方式，以及它为同一对象的另一个定义方式，因为它是为了两个不同目的而做的，然后它们冲突，你无法把这些东西重新拼凑起来。你真的需要 MathLib 库增长，才能达到可以写研究手稿的地步，真正利用综合大量不同数学领域。

So I just 2 days before I gave a colloquium in the Princeton math department where I explained in painstaking detail what Lean is, what MathLib is, but also that AI systems today—there's this MathLib library where everything is kind of perfectly done, exactly the right level of generality, and there's not supposed to be any code that's reusing code that's all—that's re-proving theorems somewhere else. Everything is the most general, the most abstract. And what AI systems are capable of doing is building like one or two layers on top of that. But what they're doing is jerry-rigging. They have not been trained yet—I'm not saying they can't do it, but they're not good at it right now—at what I call canonization, which is taking kind of sloppy code, refactoring it, out exactly where in the library it belongs, what's the canonical version of the statement of this lemma, and then every proof should just be a combination of—you apply this lemma from over here, you apply that lemma from over there, put all these things together, there's your theorem. And then it's really clean, it's really beautiful, very elegant. That's what MathLib is going for. And what I said in my colloquium is where we're at today is that you can get entire textbooks formalized as long as they're in the MathLib halo, as long as they're like one, maybe two steps removed from things that have already been done in MathLib. But when you start—and this had been my practice, my experience had been—when you start asking it to do things that are three or four steps removed from the library, the way that it came up with a definition of this object and the way that it came up with some other definition of the same object because it was doing it for two different purposes, then they clash, you can't stitch these things back together. You really need the library MathLib to grow in order to get to a place where you can do research manuscripts that are really using synthesizing tons of different areas of mathematics.

这就是为什么你会预期费马大定理需要时间，因为它远远超出 MathLib 的光环，需要相当多的时间。

Which is why you would expect Fermat's Last Theorem to take because it's so far outside of the halo of MathLib, it would take quite a bit of time.

完全正确。所以，Kevin Buzzard 有一百万美元的资助，用于未来五年，或者我想他可能已经进行了一两年，将费马大定理的形式化归约到 1980 年代已知的东西。所以，他想完成最后一步，即怀尔斯工作以及怀尔斯-泰勒的那七年，最终完成费马大定理，但建立在那些我们知道会极其困难的东西之上。他不打算做 Langlands-Tunnell，不打算做 Mazur，有各种各样的事情预计比这五年要做的更困难得多。而我在那个报告中所说的一切都是错的。我想我的报告是在周三，周五就出现了一个 1300 万行代码的发布，从公理证明了费马大定理。

Exactly right. So, Kevin Buzzard had a million-dollar grant to spend the next five years, or I think maybe he's already a year or two into that grant, to reduce the formalization of Fermat's Last Theorem to things that were known in the 1980s. So, he wanted to do that last step, that those seven years of Wiles's work and Wiles-Taylor to finally finish Fermat's Last Theorem, but built on top of things that we knew would be extraordinarily difficult. He wasn't going to do Langlands-Tunnell, he wasn't going to do Mazur, there were all kinds of things that were expected to be even much more difficult than what these five years would take. And everything I said in that colloquium was wrong. And I think my colloquium was on a Wednesday, on Friday there was a drop of 13 million lines of code from the axioms Fermat's Last Theorem is proved.

费马大定理的陈述完全是初等的，没有语义对齐需要检查，你只需看一眼，是的，那就是陈述。Lean 说它没有 sorry。

The statement of Fermat's Last Theorem is completely elementary, there's no semantic alignment to check, you just glance at it, yes, that's the statement. It's Lean says it's sorry-free.

你能解释一下它是怎么做到的吗？

Can you explain how it was done?

怎么做到的？是 Anthropic，做法就是 AI 系统变得非常非常擅长这个。它仍然不是一个漂亮的证明。它不是干净的证明，不是——要让它成为 mathlib 质量的证明，还有大量工作要做。我认为那是重要的工作，也是我们想做的。但就获得那个没有 sorry 的定理而言，他们直接冲过了所有这些。他们临时拼凑这个，临时拼凑那个。你知道，1300 万行之后，他们得到了费马大定理的一个没有 sorry 的形式化 Lean 证明。绝对令人震惊。

How it was done? So, it was Anthropic, and how it was done was AI systems just got really, really good at it. It's still not a nice proof. It's not the clean proof, it's not—there's still a ton of work that needs to be done to make that proof a mathlib quality proof. And I think that's important work and work that we want to do. But in terms of just getting that sorry-free theorem, they just blasted through all of these. They jerry-rigged this. They jerry-rigged that. And you know, 13 million lines later, they had a sorry-free formal Lean proof of Fermat's Last Theorem. Just absolutely shocking.

顺便说一句，这总是很美妙——因为一切变化太快，会有很多陈述，每个人、世界专家做出的陈述，是错误的。这就是重点。人类讨论的有趣之处就在于我们有分歧，我敢说今天所说的所有内容中，大约 40% 到 80% 在 5 年后会被证明是错的。因为一切变化太快。但这正是人类讨论、审议、话语乃至科学过程的重点：说出来，看看什么能站得住脚。

By the way, it's beautiful to always—because of how rapid everything is, there's going to be a lot of statements that everybody makes, the world experts make, that are wrong. That's the whole point of it. And that's the fun thing about human discussion is we have disagreements and I would venture to say somewhere between 40 to 80% of everything said today will be wrong in 5 years. Because everything's changing so fast. But that's the whole point of human discussion and deliberation and discourse and even the scientific process is saying and seeing what sticks.

随着时间推移。

Over time.

是的。

Yeah.

10 年、100 年，等等。

Over 10 years, over 100 years, and so on.

我在描述我的经历。我在 Lean 和用 AI 方面有很多经验。我开始做 Lean 的时候，AI 还不存在，以我们现在理解的意义上说。那非常费劲。然后 AI 系统一出现，你知道，第一批 LLM 上线。它们在 Lean 上很烂，但后来好了一点，又好了一点。我就想，‘天哪，我可以做库搜索了。我不必坐在那里找——我知道有一个引理会做这个，但我不知道它叫什么，也不知道它在流形库里具体哪个位置？是在代数几何库里吗？给我这个引理。’它很擅长这个。我就想，‘哦，这太棒了。我要继续用这些系统。’我的每一点经验都告诉我，它们能做到 X，但做不到 10X。而这是百万 X。这完全超出了我心目中系统能力的分布范围。

I was describing my experience. I had a lot of experience with Lean and with using AI. I started doing Lean before AI existed in the sense that we understand it now. And it was very painstaking. And the second AI systems came on, you know, the first LLMs were coming online. They were crappy Lean, but then they got a little bit better, a little bit better. And I was like, 'Oh my god, I can do library search. I don't have to sit there looking—I know there's a lemma that does this, but I don't know what it's called and I don't know which exactly is it in the manifolds library? Is it in the algebraic geometry library? Like give me this lemma.' It was great at that. And I was like, 'Oh, this is amazing. I'm going to keep using these systems.' And every bit of my experience told me that they can do like X, but not 10X. And this was a million X. It was just completely outside of the distribution in my mind of what systems were capable of.

你大概估计有多少人参与其中？在生成那 1300 万行代码的过程中，人类的参与程度如何？

Do you have a ballpark estimate of how many humans were involved in that? And like what is the level of human involvement in generating those 13 million lines of code?

我相信是两三个人。

I believe it was two or three humans.

好的。

Okay.

我相信他们大部分的参与就是‘继续’。全大写。

And I believe most of their involvement was 'keep going.' All caps.

好的。所以你认为真的只是温和的引导，而不是路径创建和——

Okay. All right, so you think it's really just like gentle guidance, not like pathway creating and—

据我所知，没有领域专业知识。

As far as I understand, there was no domain expertise.

好的。

Okay.

在参与做这件事的人身上。

On the parts of the people who were doing this.

就像 OpenAI 或别的什么梗那样，你知道，提示是‘解决数学’

As the meme goes with OpenAI or whatever, you know, the prompt is 'solve math'

是的。

Yeah.

之类的。

or something.

对。

Right.

做出突破。

Do a breakthrough.

是的，是的。做——是的，做出突破。

Yeah, yeah. Do—yeah, do a breakthrough.

做出突破。这就是提示。

Do a breakthrough. That's the prompt.

然后它就去做了。

And it goes and does it.

当然，你永远不知道——从公司角度，有很大的动机把它说成那样，尽管在训练的各个阶段可能有很多中间步骤，或者有些事没说，即使最终提示是‘做出突破’，在那之前可能有很多基础工作导致了——我们不知道。

Of course, you never know—from a company perspective, there's a lot of incentive to sort of sell it as that, even though there might have been a lot of intermediate steps that led at any stages of the training or maybe some things left unsaid, even though the final prompt is 'do a breakthrough,' there could be a lot of groundwork before that that led to—we don't know.

我记得那是个周五，因为我完全——我的脑子完全炸了，那是劳动节，我们刚带孩子去海滩，我需要几天时间，你知道，我不看手机，不上 Twitter，我就只是扔，你知道，把我的孩子扔进海里，然后——

So, I remember that was a Friday because I just was completely—my mind was completely exploded and it was Labor Day and we just took the kids to the beach and I needed a few days of, you know, I'm not checking my phone, I'm not on Twitter, I'm just throwing, you know, throwing my kids into the ocean and—

你有没有那样一个时刻，就坐在那里看着大海，思考这一切意味着什么？

And did you have that moment where you're just sitting and looking at the ocean and contemplating like what does it all mean?

是的。是的。

Yeah. Yeah.

所以，我一个月前已经经历过那个了。

So, I had already gone through that a month before.

是的。

Yeah.

我就想，我——所以，现在这说明自动形式化基本上解决了。

And I was like, I—so, now what that says is that autoformalization is basically solved.

是的。

Yeah.

它并不是以我希望的方式解决的。我希望——人们仍然会非常非常努力地研究这个。还有各种各样其他事情发生，就像当你证明定理时，会有各种各样其他事情作为那个过程的副产品，对人类理解而言。当你致力于构建一个美丽的数学库时，还有各种各样其他事情。那个过程会继续。那个过程会非常困难。它将是，你知道，世界各地的人们以这种开源方式贡献的数千个工时。但就你能得到一个人类写在纸上的东西的没有 sorry 的证明而言，我认为那个问题已经完成了。

It's not solved in the way that I would want it to be solved. I want—and people are still going to work very, very hard on this. There's all kinds of other things that happen, just like there's all kinds of other things that happen when you prove theorems that are byproducts of that process for human understanding. There's all kinds of other things when you work on making a beautiful library of mathematics. And that process will continue. That process will be very difficult. It's going to be the, you know, thousands of man hours of people around the world contributing in this open source way. But in terms of can you get a sorry-free proof of something that human beings have written down on paper, I think that problem is done.

所以你待在那里，喝着威士忌，一个月前经历过这个，现在又经历一次。

And so you're staying there with a whiskey having gone through this a month before and now again.

对。这是——

Right. This was—

第三轮。这是第——

Round three. This is round—

因为劳动节回来时是纳维-斯托克斯。

Cuz when I got back from Labor Day it was Navier-Stokes.

纳维-斯托克斯。但这次——

Navier-Stokes. But this one—

只是在哀悼，如果 FLT 你要这样对我？拜托。

was just mourning if FLT you're going to do this to me? Come on.

但现在，我是说这真的触动了，对吧？随着准黎曼假设。

But now, I mean this really hit home, right? With the quasi-RH.

是的。

Yeah.

那么也许你能谈谈素数的重要性和美妙之处，以及从准黎曼假设到完整黎曼假设的飞跃有多大、多难？

So maybe can you speak to the importance and the beauty of the primes and also how big and hard the leap is from quasi-RH to full RH?

是的，所以素数，素数之所以如此有价值，是因为它们是乘法理论中的基本粒子。每个整数都分解为素数的乘积。你知道，我们知道物理学中的基本粒子，所有物质都由它们构成。同样，素数是构成整数乘法结构的粒子。它们的分布一直是个谜，现在仍然是谜。即使黎曼假设也不能解决我们想知道的关于素数的一切。但黎曼假设如此美丽，因为它指明了分布应该是什么样，并且它在非常遥远的领域有上千种应用。只要黎曼假设是真的，我们就会知道 XYZ。

Yeah, so the primes, the reason the primes are so valuable is that they are the fundamental particles in the theory of multiplication. Every whole number is broken down into a product of primes. You know, we know about the elementary particles in physics, and all of matter is made up of that. In the same way, the primes are the particles that make up the multiplicative structure of whole numbers. And their distribution had been a mystery and remains a mystery. Even the Riemann hypothesis doesn't solve everything we want to know about prime numbers. But the Riemann hypothesis was so beautiful because it pointed the way to what the distribution should be and it has a thousand applications in very far away fields. If only the Riemann hypothesis were true, then we would know XYZ.

让我插一个小故事，是上周发生的，挺有意思的。Grant Sanderson 访问了 IAS，我们进行了一次长讨论，关于 AI 正在对数学界做什么。我非常喜欢 Camillo De Lellis 的评论，那是——我会尽力模仿。我模仿不了口音。

So let me interject a little story that came up last week, which was pretty entertaining. So we had Grant Sanderson was visiting IAS and we had a long discussion about what is AI doing to the math community. And I really enjoyed this comment that Camillo De Lellis made, which was—I'll do my best impression. I can't do the accent.

是的。但有一位数学家在 20 世纪 50 年代写过一个剧本。谁知道数学家还会写剧本呢，但显然这个剧本的前提是冯·诺依曼死了，上了天堂。他见到了上帝，并且有机会问一个问题。于是他问：“黎曼猜想是真的吗？”当然，大家都屏住呼吸等待答案。上帝说：“是的。”终于大家可以松口气了，然后冯·诺依曼问：“但你怎么知道？”上帝想了想说：“嗯，我知道所有零点的位置，它们都在 R 等于二分之一那条线上。”所以我的理解是，这个剧本的前提是，这对冯·诺依曼来说还不够。他有点激动，想进一步了解为什么它是真的，我想他最后去了另一个地方。所以我不知道从中能得出什么道德教训，但是嗯

Yeah. But there was a play written in the 1950s by a mathematician. So who knows that mathematicians are writing plays, but apparently this play, the premise of it was that von Neumann died and he went to heaven. And he met God and he got to ask him a question. So he asked him, "Is the Riemann hypothesis true?" And of course everyone holds their breath as they're waiting to hear the answer. And God says, "Yes." So finally everyone can exhale and then von Neumann asks, "But how do you know?" And God thinks and says, "Well, I know where all the zeros are and they're all in the R equals a half line." So my understanding is the premise of the play is that that's not enough for von Neumann. He gets a little bit agitated to try to find out more about why it's true and I think he ends up at the end in the other place. So I don't know what draws a moral from it, but um

是的。

Yeah.

这似乎道出了数学体验的本质。

It seems to speak to the mathematical experience.

数学不是事实，而是“为什么”。它一直是“为什么”。这是我们最深层的对理解的追求。如果你只是告诉我一个事实是真的，但我不理解为什么它是真的，那对我的知识贡献为零。所以我认为素数也很有趣，因为它们与三岁小孩能理解的东西相关。比如我们从根本上知道如何给数字加一。你知道，小时候有一个突破性的时刻：“我可以永远加一。”这是一个重大的想法。我可以永远加一。然后你学会了乘法和分组，这也是一个重大的想法。素数对我来说很有趣，而且一直很有趣，因为我们不理解加法和乘法如何协同工作。数学中两个最基础、最简单的基本概念，我们不知道它们如何协同工作。我可以给一个我能因式分解的数加一，但我完全不知道会发生什么。我知道如果它是偶数就会变成奇数，反之亦然，但我不知道一个有很多很多复合因子的数是否会变成素数。我们从根本上不知道那个简单的，你知道，加一如何改变乘法。而素数基本上就是那个三岁小孩之谜的提炼。这是一个我们所有人仍在努力解决的谜。你知道，有趣的部分在于，如果证明告诉了我关于那个结构的一些根本性的东西，让我能够实际使用。

Math is not the fact, it's the why. It's always been the why. It's our deepest quest for understanding. And if you just tell me a fact is true, but I don't understand why it's true, that has contributed zero to my knowledge. So primes I think are also interesting because they relate to things that a three-year-old can understand. Like fundamentally we know how to add one to numbers. And you know, there's this breakthrough moment when you're a kid, "I can add one forever." That's a big a big idea. I can keep adding one forever. Then you learn how to multiply and group and that's also a big idea. And primes are interesting to me and have always been interesting because we don't understand how adding and multiplying work together. The two most fundamental basic simple ideas in all of mathematics, we don't know how they work together. I can add one to a number that I can factor and I have no idea what happens. I know if it's even it becomes odd and vice versa, but I don't know if a number that has many, many composites becomes prime. We just fundamentally don't know that simple, you know, adding one, how that changes multiplication. And primes are basically, you know, this like the the distillation of that three-year-old mystery. And it's it's a mystery that all of us still wrestle with. You know, that would be the the interesting bit as if the proof told me something fundamental about that structure that I could actually use.

是的，对那个三岁小孩来说，我想我的大孩子当时大概两三岁。他在玩我们有的那些小积木，你知道，孩子们喜欢玩的立方体。

Yeah, to to the the three-year-old, I think my my older one was something like two or three. He was playing with We had these little blocks, you know, cubes that kids like to play with.

是的。

Yeah.

嗯，一包大概有 20 个。所以他会摆一个 4×5 的小网格，他喜欢那样，还会摆一个 10×2 的网格，那挺酷的。然后有一天我看过去，他非常沮丧。我问：“怎么了？”他说：“你知道，这不行。”其中一个滚到沙发下面去了。

And uh there was like 20 of them in in the pack. And so he would make like a little 4 by 5 grid, and he liked that, and he would make a 10 by uh two grid, and that was kind of cool. And then one day I look over, and he's like really frustrated. Like, "What's going on?" He's like, "You know, it's not it's not working." And one of them had rolled under the couch.

是的。

Yeah.

他只有 19 个，他想不出办法用这 19 个摆出一个好看的矩形。

And he only had 19, and he couldn't figure out a way to make a nice-looking rectangle out of the 19.

是的。这至今仍困扰着我。

Yep. That still bothers me today.

是的。所以，绝对地，孩子们喜欢素数。孩子们能发现素数。这

yeah. So, absolutely kids love primes. Kids can discover primes. It's

所以解决办法是再丢一个。

And so the solution is to lose another.

是的。

Yeah.

嗯，正是如此。是的。是的。

Well, exactly. Yeah. Yeah.

你认为，从准黎曼猜想到黎曼猜想之间的跨越，我猜你认为人类在你的有生之年，没有 AI 的帮助，无法解决完整的黎曼猜想？

Do you think uh so the the leap between quasi and RH, I presume you think humans will not in your lifetime be able to solve without AI's help the full Riemann hypothesis?

我不知道。AI 能不能，我猜，是接下来的问题。比如，它有多难？

I don't know. Can AI, I guess, is the following question. Like, how hard is it?

我不知道。我不知道。我没想到我会看到准黎曼猜想的证明。现在我看到证明是什么了，还有很多技术细节我仍需为了自己满意而仔细研究。有一个 Lean 形式化。Lean 陈述已经在库里了。已经在 mathlib 里了。所以，他们有一个没有 sorry 的证明，这让我 100% 确信它是对的。而且，读了证明之后，你知道，在高层次上，它使用的想法都是完全有效的。它们很精彩，你知道，嵌入由六次特征扭曲的莫比乌斯函数。这些都是解析数论学家家常便饭的东西，被以恰到好处的方式串联起来，绝对令人难以置信。

I don't know. I don't know. I did not think I would see a proof of quasi RH. Now that I see what the proof is, and there are a lot of technical details that I still have to go through for my own uh satisfaction. There's a lean formalization. The lean statement is was already in the library. Was already in mathlib. So, the fact that they have a sorry-free proof gives me 100% confidence that the the is right. And moreover, having read the proof, you know, at a high level, the ideas that it's using are all completely valid. They're brilliant, you know, to embed the Mobius function twisted by sextic characters. This is all stuff that's the bread and butter of analytic number theorists that was chained in just the right way and was absolutely incredible.

那么，黎曼表明的是，嗯，记住我们讨论的是这个函数，它有复数输入，有这些输出，我们关心的是输出何时为零。所以他表明存在所谓的临界带，这是复平面上的一个垂直无限带，实部，你可以认为 X 坐标在 0 和 1 之间，虚部，Y 坐标，可以任意高或任意低。而猜想是零点都位于那个带的正中央。实部为二分之一。这正是 Ankur 刚才说的。而直到周二我们所知道的是 zeta 函数没有零点的区域。这个区域随着你往上走会越来越窄。它趋向于那条线。那条线是所谓素数定理，它告诉你素数的基本分布等价于这个函数在实部 S 等于 1 的那条线上没有零点。所以，如果 X 坐标是 1，当你沿着那条线上下走，你永远不会找到应用 zeta 函数后它为零的值。明白吗？所以，我们有一个无零区域，随着你往上走越来越窄。这给了我们已知最好的素数分布，素数定理中最好的误差项。如果你告诉我你想讨论多少数学，我可以深入讲。

And so, what Riemann showed is that, uh, remember we're talking about this function that has complex inputs and has, uh, these outputs and we're worried about when the output is zero. So, he showed that there's what's called the critical strip, which is, uh, vertical infinite strip in the complex plane where the real part is, so you can think about the X coordinate lies between zero and one, and the imaginary part, the Y coordinate, can go arbitrarily high or arbitrarily low. And the hypothesis is that the zeros all lie right in the dead center of that strip. Where the real part is a half. That's exactly what Ankur was just saying. And what we knew until Tuesday was region where there are no zeros of the zeta function. That's like getting thinner and thinner the higher up you go. It goes towards the one line. The The one line is something called the prime number theorem, which tells you the basic distribution of the prime numbers is equivalent to the fact that there are no zeros of this function on the line, uh, real part of S equals one. So, if the X coordinate is one, as you go up and down that line, you will never find, uh, value where when you apply the zeta function, it it hits zero. Okay? So, we had the zero-free region that was a that was getting thinner and thinner the higher up you go. And that gave us the best known distribution on the prime numbers, the best error term in the prime number theorem. Which I can go into if you You tell me how much math you want to You want to talk about.

来吧。

Let's go.

但是，我本可以预料到的是，它会稍微加厚那条线。嗯，随着你，你知道，它仍然会趋向于零。厚度仍然会随着你沿虚轴向上而趋向于零，但是，嗯，会比我们目前知道的更好。那个厚度 70 年没变过，但我可以想象它或许能取得一些进展。它做的是给了我们一个垂直带。所以，实部在 7/8 和 1 之间没有零点。一个完整的无限垂直带。这就是准黎曼猜想。如果你能把那个 7/8 降到二分之一，那就是完整的黎曼猜想。现在，它在 7/8 证明中使用的方法，还有第二个证明大约是 11/12，别管这些技术细节。那个方法从根本上无法降到 1/2。

But, what I could have anticipated is that it would thicken that line a little bit. Uh, as you, you know, it would still be going to zero. The The thickness would still be going to zero as you go up the, uh, imaginary axis, but, uh, there would be better than than what we currently know. That thickness hasn't moved in 70 years, but I could sort of imagine that it could maybe make some progress. What it did is it gave us a vertical strip. So, there are no zeros with real part between 7/8 and and 1. An entire infinite vertical strip. That's a quasi Riemann hypothesis. And if you can get that 7/8 down to a half, that's the full Riemann hypothesis. Now, the methods that it used in the 7/8 proof, and there's a second proof that's about 11/12, never mind these technical details. That method sort of fundamentally can't get down to 1/2.

所以，需要新的想法。

So, a new idea is needed.

一个新的想法，但准黎曼猜想需要很多很多新想法，而且它想出来了。

a new idea, but quasi RH needed lots and lots of new ideas, and it came up with them.

既然你们已经经历过三次了，我想问你们三位：你们对各类问题难度的直觉是否需要重新校准？因为你提到挑简单的事情做。想必你们有很好的难度雷达。比如这个问题可能可解，那个今天不可解、这周可解、这个月可解。你们是否在重新适应这个新世界？

As you've gone through this three times already, is your intuition—this is for all three of you—about remeasuring in your mind the hardness of the various problems? Because you mentioned picking easy things. Presumably you have a good hardness radar. Like this problem is probably solvable and this is not solvable today, solvable this week, solvable this month. Are you readjusting to this new world you're in?

我在重新调整我无法预测它的能力。

I'm readjusting my inability to predict it.

是的。

Yes.

没错。

Exactly.

可你本来有这能力，不是吗？只是噪声大。

Well, you had the ability, no? It's noisy.

当然。噪声比我以为的大得多。我的意思是，我们一直知道有些我们关心的问题在现有工具的能力范围内。但令人惊讶的是规模——有多少不同的问题真的属于这一类。所以我们也还有很多不知道的——显然他们提示了数千个问题，而他们只公布了成功的解。成功率只有大约 10%。所以我们只能猜测：是否存在一个非常参差不齐的边界，我们可以眯着眼说能理解它，区分哪些成功了、哪些没有？我认为另一个危险是，也许最安全的问题恰恰是那些你一看就痛苦地明显需要真正有创意的新想法的问题。因为你能感觉到也许存在一个障碍，而文献中现有的东西似乎就是填不上那个洞。但现在我认为永远改变数学格局的一件事是：也许人们对这些想法会比过去更保密。因为一旦这些想法公开，AI 就可能更容易补上那一个新想法、那个缺失的成分、它需要的那块新拼图，并能够完全穷尽地探索它能从中达到什么。所以你可能会比过去更保护自己的想法。我觉得这也有点可怕，因为传统上数学界一直非常开放地分享想法，失去这一点会是件坏事。

Sure. It's a larger amount of noise than I thought. I mean, I think we've always known that there are problems we care about that are within reach of the tools that are out there right now. But it's just surprising the scale at which—how many different problems really fall into that category. So, we also don't know a lot about—apparently they prompted thousands of problems and what they dropped were only the positive solutions. That they only had about a 10% success rate. So, we're sort of left guessing: is there a very jagged boundary that we could squint at and say we can make sense out of that distinguishes what did fall versus what didn't fall? I think one of the other dangers is that maybe the problems which are the safest are ones which you look at and it's just painfully obvious you need a really creative new idea. Because you can sort of tell that maybe there's a barrier and the things that are out there in the literature just don't seem to fit into that hole. But now I think one of the things that's changed the landscape of math forever is: maybe people become a little bit more secretive with those ideas than they were in the past. Because once those ideas are out there, then it might be a lot more straightforward for AI to be able to balance that one new idea, that missing ingredient, that new tile it needs and be able to fully exhaustively explore what it can reach out of that. So you might be a lot more protective of your ideas than you were in the past. I think that's also a bit scary because traditionally the mathematics community has been really open with sharing ideas and that would be something bad to lose.

嗯，那也是因为我们有这种先到先得的心态，我认为那也会改变。

Well, that's also because we had this first-through-the-gate mentality and I think that's also going to change.

没错。

That's right.

我更希望开放保留下来，而先到先得的心态消失。

I would prefer that the openness stay and the first-through-the-gate is what goes.

但那需要时间。

But that'll take time.

需要时间。

It'll take time.

是的。所以那可能是一种更短暂的影响，或者如果我们能学会如何重视其他东西，那也许就会消失。其他社群有如何评估贡献的模式，即使不把某个人的贡献捧上神坛。所以并非没有前进的道路，但短期内这确实是我们像过去那样做数学、像过去那样享受数学的真正障碍。

Yeah. So that could be an effect that's more transient in nature or that if we can learn how to value other things, then maybe that'll go away. There are other communities that have models for how to value contribution even when it's not lionizing one individual's contribution. So it's not that there's not a way forward, but that is in the short term a real roadblock to doing mathematics the way that we used to and enjoying it the same way that we used to.

说到这个，Steve，你给我们讲过一个关于汤的故事，可能让你爱上了流体动力学这个更大的问题。所以，从数学到应用数学再到工程，整个链条。也许你能谈谈纳维-斯托克斯方程，这套解释流体的方程，为什么它如此重要，为什么它美丽。就像我们谈论素数一样，为什么它如此困难，以及纳维-斯托克斯被证明为什么是件大事。

So speaking of which, Steve, you told us a story about soup that made you maybe fall in love with the bigger problem of fluid dynamics. So, going from math to applied math to engineering, the full thing. So maybe can you talk about Navier-Stokes, these set of equations that explain fluids, why it's such a big deal, why it's beautiful. Similar as we talked about primes, why is it so hard and why is it such a big deal that Navier-Stokes was proven.

从根本上说，纳维-斯托克斯方程只是陈述流体动量守恒的一种方式，适用于气体和液体这类可以相对自由运动的物体。我们知道 F=MA 是动量守恒，经典牛顿物理学。同样的基本原理适用于这个房间里的气体、流动的水，并且在任何体积中都成立。所以它在小体积、大体积、假想体积中都成立。如果你在一个体积中写下动量守恒，然后取该体积任意小的极限，就得到这些纳维-斯托克斯方程。它是一个微分方程，一个偏微分方程，描述流体的速度场如何在空间和时间中变化，以及与该流体所受作用力相关的压力。

So, fundamentally the Navier-Stokes equations is just a way of stating the conservation of momentum for fluids, for objects that can move kind of freely like gases and liquids. So, we know F equals MA is conservation of momentum, classical Newtonian physics. The same basic principle applies to the gas in this room, to water flowing, and it's true in any volume. So, it's true in little volumes, big volumes, kind of imaginary volumes. And if you write down that conservation of momentum in a volume, and then you take the limit as that volume becomes arbitrarily small, you get these Navier-Stokes equations. It's a differential equation, a partial differential equation, that describes how the velocity field of a fluid changes in space and time, and also the pressure related to the forces acting on that fluid.

你能直观解释一下吗？既然 F=MA 是个相当简单的方程，为什么流体是这么复杂的系统？

Can you give intuition about why fluid, given that F equals MA is a pretty simple equation, why fluids are such complicated systems?

F=MA 写起来简单，但它可以变得非常非常快就复杂起来。比如，如果我有单摆，写出单摆的方程很容易。如果我在它末端再挂一个摆，现在方程就更复杂了，而且会产生混沌。流体在某种意义上，如果我要模拟这个房间里的空气，我可以把房间切成十亿个小盒子，为这十亿个小盒子中的动量变化写下微分方程，但现在我得到了一个非常复杂的方程组。所以，它写起来简单，概念上理解方程在说什么、代表什么也很简单，但真正求解它、以任何实际意义理解流体如何运动，是非常非常复杂的。

F equals MA is simple to write down, but it can get complicated really, really quickly. So, if I have like a single pendulum, it's really easy to write down the equations for that single pendulum. If I put another pendulum on the end of it, now I have a more complicated equation, and it gives chaos. Fluids is, in a sense, if I try to model the air in this room, I could chop the room up into a billion little boxes, and write down a little differential equation for how the momentum works in all of those billion little boxes, but now I have a really complicated system of equations. And so, it's simple to write down, it's simple to conceptually understand what the equation is saying and representing, but actually solving it and understanding how the fluid actually moves in any kind of practical sense is very, very complicated.

我读到过——流体动力学重要的一个领域是减少飞机飞行时的阻力。我读到航空公司通过减少 1% 的阻力可以节省数百亿美元。那么，为什么这么难？这是一个。而阻力可能比湍流之类更容易。那么，为什么这些价值如此之高的事情如此困难？

I read something like—one space where fluid dynamics is important is reducing drag for when you're flying for airplanes. And I read something like airlines could save something like tens of billions of dollars by reducing the drag by 1%. So, why is this so hard? So, that's one. And drag is maybe easier than something like turbulence. So, why are these things that are worth so much money so difficult?

好消息是，如果你能提高我们预测湍流的能力，哪怕只是稍微更好、更快、更准确地模拟它，有几十个领域可以通过相对较小的改进产生巨大影响——包括经济、生态、人类健康和安全。所以，减少飞机阻力、更好的人工心脏、更高效的风力涡轮机和燃气轮机，等等。世界是——我们生活和工作在流体中，我们所有的机器也是如此。逆设计问题。我知道它实际上与这类数学问题密切相关。我知道我想证明什么，但我不知道如何到达那里。我知道我想要一个高升力、低阻力的机翼，但我不知道改变几何形状、改变机翼形状会如何影响这一连串越来越小的涡流——在机翼上产生的湍流流体——最终导致其升力和阻力。

So, the good news is that if you can improve our ability to predict turbulence, to simulate it even a little bit better, faster, more accurately, there are dozens of fields where you can have a huge impact—both economical, ecological, human health, and safety—with a relatively small improvement. So, drag reduction on airplanes, better artificial hearts, more efficient wind turbines and gas turbines and, you name it. The world is—we live and work in a fluid, and so do all of our machines. The inverse design problem. I know it's actually very closely related to this kind of math problem. I know what I want to prove. I don't know how to get there. I know I want a wing that has high lift and low drag, but I don't know how changing the geometry, changing the shape of that wing, will affect this whole cascade of smaller and smaller and smaller vortices—this turbulent fluid that generates over the wing—that eventually results in its lift and drag.

我不知道如何高效地做逆向设计。部分原因是这些方程模拟或测试起来很昂贵。风洞很贵，你需要超级计算机来模拟这些东西。这其实和纯数学有点关系——我想我们待会儿会回到这个话题。但费曼有一句非常好的话：我们知道如何写下这些非常通用的方程——麦克斯韦方程、纳维-斯托克斯方程、薛定谔方程——但我们目前无法从方程本身知道哪些现象是可能的。如果你从未到过地球，你能仅凭一个偏微分方程就预测出日落、潮汐和所有这些现象吗？我们离那还差得远。而且还有一些非常根本的数学问题：如果我能写下一个微分方程，我却无法告诉你它里面包含什么。我们仍然必须模拟，仍然必须测试，仍然必须走这套非常繁复的工程流程，才能理解这些方程如何表现为机翼上的阻力，或人工心脏内部的湍流。

I don't know how to inverse design efficiently. Part of it is because the equations are expensive to simulate or to test. Wind tunnels are expensive. You need supercomputers to simulate these things. This is actually kind of related to pure math—we'll get back to this, I think. But there's this really nice Feynman quote: we know how to write down these very general equations—Maxwell's equations, Navier-Stokes equations, Schrödinger's equations—but we currently don't know from the equation what phenomena are possible. If you never visited Earth, could you have possibly predicted the sunsets and the tides and all the phenomena just from a partial differential equation? We're not even close to there. And there are some very fundamental math questions about—if I can write down a differential equation, I can't tell you what's inside of it. And we still have to simulate, we still have to test, we still have to do this very elaborate engineering process to understand how these equations manifest drag on an airplane wing or turbulence inside of an artificial heart.

抱歉，我能就这一点问个问题吗？我一直想问。比如在爱因斯坦的广义相对论中，他写下了这些方程。几年后，施瓦西发现了一个爆炸（奇点）。大家就说：“哦，得了吧。这些方程到底哪里有问题，才会出现爆炸？如果你把足够多的物质塞进足够小的空间，光就无法逃逸。我们没有任何物理经验支持这个。这一定是方程的错误。”直到我们发现了黑洞。

Sorry, can I just ask a question on that? I've always wanted to. So, like in the case of Einstein's theory of general relativity, he writes down these equations. A few years later, Schwarzschild finds a blowup. And they're like, "Oh, come on. So what's wrong with these equations that there's a blowup? If you get enough matter into a small enough space, then light can't escape it. We don't have any physical experience for this. This has got to be a mistake in the equations." Until we discover black holes.

对。

Right.

然后结果发现，这个理论完美地解释了——它预测了应该发生什么。而在纳维-斯托克斯方程的爆炸问题上，我们是要发现关于水的某种新东西，还是说在某个尺度上你会到达粒子层面，到达一种不再遵循液体方程的原子结构？为什么我们会预测一个允许被推到极限的连续过程应该有解呢？

And then it turns out that actually the theory perfectly explains—it predicts what's supposed to happen. In the case of the blowup of Navier-Stokes, are we going to discover something new about water, or is it the fact that at some scale you get to particles, you get to an atomic structure that does not follow the liquid equations anymore? Why would we have predicted that there should be solutions to a continuous process that's allowed to be pushed to its limits?

是的，这是个很好的问题。我给你一个工程上的回答：这些方程在某个尺度上是对流体如何运作的极佳模型。你说得完全正确——你需要在一个体积内对一定数量的单个分子取平均，这些本构方程才成立。所以如果你放大到足够小的尺度，现在我们谈论的是单个分子，纳维-斯托克斯方程就失效了。所以这些方程爆炸的数学问题从数学角度看很有趣。等到它们爆炸时，它已经不再是流体了。

Yeah, I mean, it's a great question. I'm going to give you the engineering answer, which is these equations are an exceptionally good model of how fluids work at a certain scale. And you're absolutely right—you need some number of individual molecules to be averaged over in a volume for these to be constitutive equations that are valid. And so if you zoom in small enough, now we're talking about individual molecules and Navier-Stokes breaks down. So the mathematical problem of these equations blowing up is interesting from a mathematical standpoint. By the time they blow up, it stops being a fluid.

对。

Right.

就像其他——这里没有黑洞。

Like other—there's no black holes.

没有黑洞。就像其他物理接管了。是的。

There's no black holes. Like other physics takes over. Yeah.

但有可能——等等。我觉得从科学哲学的角度看很有意思：方程失效的地方可能，也许统计上很可能，是一扇窗、一扇门，通向需要某种其他发现的突破。就像牛顿物理失效需要爱因斯坦。

But it's possible—wait. I think it's interesting from a philosophy of science perspective that where equations break down could, and maybe with statistically high likelihood is, a window, a door into a breakthrough of needing some other discovery. Like Newtonian physics breaking down needs Einstein.

对。

Right.

而大统一的问题在于，我们还没有在与亚原子粒子相互作用相同的尺度上拥有引力。

And the grand unification is that we don't have gravity at the same scales as we have interactions of subatomic particles.

但这个想法——我们对这些方程还有如此根本的东西不知道——现在地球上有那么多人在每个能想象到的行业里模拟和测试纳维-斯托克斯方程。它大概是整个工程界被研究、分析和使用得最多的方程之一。而我们却不知道这个偏微分方程的根本性质：它是否在有限时间内爆炸？这还只是更大问题的第零阶版本：它还允许哪些其他类型的解？是否存在日落、飓风、龙卷风以及所有可能存在的现象？存在性和唯一性只是最简单的婴儿学步。而我觉得有趣的地方在于：即使这个数学结果在工程意义上没有直接 tangible 的用途，它也让我们离理解这些方程的本质和可能性更近一步，这对我们来说是非常根本地有趣的。

But this idea that we don't know something so fundamental about these equations—there are so many people on Earth right now simulating and testing Navier-Stokes in every industry imaginable. It's probably one of the most studied and analyzed and used equations in the entire engineering world. And we don't know this fundamental property of the PDE: does it blow up in finite time or not? That's like the zeroth-order version of the bigger question: what other kinds of solutions does it admit? Are there sunsets and hurricanes and tornadoes and all the phenomenology that could exist? Existence and uniqueness is like the simplest baby step. And that's where it gets interesting to me: even if there's not a direct tangible use of this mathematical result in an engineering sense, it gets us one step closer to understanding the nature of these equations and what is possible, which is very fundamentally interesting to us.

这个带有爆炸的解能否以某种方式用于工程应用？你想把它驱动到这种状态，得到某种奇怪的行为，帮助飞机以更小的阻力飞行之类的？

Could this solution that has the blow-up be used somehow in an engineering application? You want to drive it into this state and get some funny behavior that helps airplanes fly with less drag or something?

是的，我认为原则上可以。一个巨大的产业，我们试图做的很大一部分就是操纵流体流动——让它们服从我们的意志，对吧？所以如果你观察鸟或昆虫飞行，它们在塑造涡旋，小心地把能量汇聚到结构中，以实现它们日常生活的目标，比如停在岩石上或捕猎。我们也试图做同样的事。所以流动控制是最高挑战之一。它非常非常具有挑战性。但如果我们有更好的能力控制这些流动，我们所有的科幻技术都会某种程度上加速。而这个证明在某种意义上就是这种非常专门的流动控制的一个非常有趣的构造。

Yeah, I mean, I think in principle. A huge industry, a huge part of what we try to do is manipulate fluid flows—bend them into our will, right? So if you watch a bird fly or an insect fly, they are shaping vortices and carefully funneling energy into structures for their objectives of their daily life, to perch on a rock or hunt. And we try to do the same thing. So flow control is one of the highest challenges. It's very, very challenging. But all of our science fiction technology would be kind of accelerated if we had better ability to control these flows. And then this proof is one very interesting construction of this kind of very specialized flow control in a way.

我想起费曼在 70 年代被问及智能、人工智能。

I'm reminded that Feynman was asked about intelligence, artificial intelligence, in the '70s.

是的。

Yeah.

他说了类似这样的话：“我们用技术解决问题的方式与自然解决这些问题的方式截然不同。看看鸟，鸟怎么飞，飞机怎么飞。”他正是指向这一点。现在回想那句话很有意思，当我们看到 LLM 在定理证明上能做什么，对比人脑在定理证明和各种其他应用上能做什么。但有趣的是，他恰好指出了这个问题：我们如何用技术解决飞行，以及这与自然的解决方案有多么不同。我们的飞机不扇翅膀。

And he said something like, "The way that we solve problems with technology is radically different from how nature solves these problems. Look at birds, how birds fly and how airplanes fly." He was exactly pointing to this. And it's kind of interesting to think back to that comment now when we see what LLMs can do with theorem proving versus what human brains can do with theorem proving and all kinds of other applications. But it's interesting that he pointed exactly to this issue of how we've solved flight using technology, and how different that is from nature's solution to it. Our airplanes don't flap.

现在我想知道，当我看到飞机飞过时，鸟是否会有存在危机。

Now I wonder if birds have existential crises when I see planes flying.

正是。

Exactly.

我到底为什么在这里？

What am I even here for?

是啊。是啊。为了抓虫子。

Yeah. Yeah. To get the worm.

纳维-斯托克斯方程有哪些不同的组成部分？你能深入讲讲吗？描述流体需要什么？

What are the different components of the Navier-Stokes equation? Can you dig in? What does it take to describe fluid?

好，非常非常基础：流体在运动。它在不同地点和不同时间以不同方式运动。所以流体的基本构件是速度场。在空间中的每一点都有一个速度矢量，告诉你流体如何运动。也许这里很快，那里很慢，就形成了一个涡旋。

Yeah, so very, very basic: a fluid moves. And it moves differently in different places and at different points in time. And so the basic building block of a fluid is the velocity field. So at every point in space there is a velocity vector that tells you how that fluid is moving. Maybe it's really fast here and slow here, and it creates a vortex.

当流体速度在空间上变化很大时，事情就变得更有趣了。比如龙卷风就是一个在空间和时间上都变化非常快的流体速度场。其中涉及压力。龙卷风的核心是低压。所以压力在推动这个速度场，使其以某种方式运动。基本构件是速度场和压力。然后方程告诉你速度如何根据自身的动量在时间和空间上变化。流体有动量来携带自身，这就是阵风。它有内部摩擦和阻力，这就是黏度。蜂蜜比水有更大的内部阻力，水比空气有更大的内部阻力。然后所有外力，比如加热、压力和重力等。这些都可以量化为这个更大的纳维-斯托克斯方程中的项。

Um and it gets more interesting when the fluid velocity changes a lot over space. So like a tornado is a very rapidly changing fluid velocity field in both space and time. Um there's pressures that are involved. So again, like the core of a tornado is low pressure. So pressures are um kind of forcing this velocity field and causing it to move in the way it does. That's the building block is a velocity field and its pressures. And then the equations tell you how that velocity changes in time and in space according to kind of its own momentum. So fluid has momentum to carry itself. That's, you know, gusts of wind. It has internal friction and resistance. That's viscosity. So honey has more internal resistance than water, has more internal resistance than air. Um and then all of the external forces like um heating and pressure and gravity and things like that. Um so each of those can be quantified as a term in this this larger Navier-Stokes equation.

说到外力，你能描述一下纳维-斯托克斯方程数学方面的爆炸吗？问题的表述是什么，实际证明了什么，为什么这么难？

So speaking of external forces, can you describe the math side of this of the Navier-Stokes with a blow up? What was the formulation of the problem and what was actually proven and why was it so hard?

如果没有内部摩擦，即黏度，我们强烈怀疑方程在某些条件下会爆炸。我们有其他例子，更简单的问题类似于纳维-斯托克斯方程，确实会爆炸。最简单的就是一个简单的常微分方程，如果你不介意我稍微跑题一下。

If you don't have that internal friction, that viscosity, we have a strong suspicion that the equations can and will blow up under certain uh conditions. We have other examples, kind of simpler problems that are like Navier-Stokes, uh that do blow up. So the simplest one is just a simple ordinary differential equation, if you don't mind me taking a little tangent.

请讲。

Just go.

所以 x. = x²。x 是系统的状态。x. 表示 x 随时间的变化率。如果 x 的变化率与 x² 成正比增长，那个微分方程确实会在有限时间内爆炸。所以只需要一个二次非线性项。想想如果我有 x. = x。这就像 x 可以是兔子的数量。我们知道兔子喜欢繁殖。x. = x 呈指数增长。兔子数量随时间呈指数增长，会变得越来越大，翻倍再翻倍。但它不会在有限时间内达到无穷大，那只是正常的指数增长，不是有限时间爆炸。那是 x. = x。但如果我取 x. = x²，那就会在有限时间内爆炸。所以二次非线性，这种二次反馈足以导致方程爆炸。如果你写下这个，你可以用经典的基本微积分技巧求解，你会得到有限时间爆炸。所以有一系列偏微分方程具有相同的行为，其中二次非线性项。有趣的是，纳维-斯托克斯方程中描述流体如何凭借自身动量运动的对流项就是一个二次非线性项，就像 x. = x² 一样。这是自运动。没有扩散，没有黏度，我们认为那可能会在有限时间内爆炸到无穷大。但我们不知道黏度，即像蜂蜜那样的内部摩擦和阻力，是否总是能在爆炸发生前将其阻尼掉。作为工程师，我的直觉是当然会。如果有一个加速到无穷大的流动，它应该立即扩散到邻近区域，产生扩散并抑制增长。所以我认为数学问题本质上是：是否存在任何边缘情况，任何流体的初始配置，使得你能完美地编排，使其在有限时间内加速到无穷大。然后有两个技术版本：一个没有外部强迫，一个有光滑外部强迫。光滑很重要，因为如果外部强迫有奇点，那就变得平凡了。比如，如果我用龙卷风强迫，一个无限紧的龙卷风，那当然会爆炸。

So x. = x². x is the state of my system. x. just means the rate of change of x in time. And if that rate of change of x is is growing proportional to x² that differential equation does blow up in finite time. So it just takes a a quadratic nonlinearity. Um think about if I have, you know, x. = x. This is like x could be the population of bunnies. And we know bunnies like to populate. x. = x grows exponentially. You know, the the state of my bunny population in time will grow exponentially. It'll get bigger and bigger and bigger. It'll double and double and double. But it won't reach infinity until infinite time is past. That's just a normal exponential growth. That's not finite time blow up. That's x. = x. But if I take x. = x², that blows up in finite time. So that quadratic nonlinearity, that quadratic kind of feedback is enough to cause this equation to blow up. And you can like if if you write this down, um you know, you can solve this using kind of classic basic, uh, calculus techniques and you get this this this finite time blow up. Um, and so there's a progression of partial differential equations that have the same behavior where a quadratic non-linearity and that's really interesting is the Navier-Stokes equations one of the terms that can the convective term that tells how the fluid kind of moves itself with its own momentum is a quadratic non-linearity just like x. = x squared. It's kind of this self-motion. Um, without diffusion, without viscosity we think that that should probably blow up, uh, to infinity in finite time, have this finite time blow up. But we don't know if viscosity, if the kind of, you know, internal friction and resistance like honey has always damps out those blow ups before they can happen. My intuition as an engineer is, of course they do. Like if I have a a flow where it's you know, accelerating, uh, you know, to infinity, that should immediately diffuse into its neighbors. It should create this like diffusion into its its neighboring fluid and kind of tamp down that that growth. And so I think the math problem is essentially are there any edge cases, any initial configurations of the fluid where you perfectly orchestrate this thing so that it could actually accelerate to infinity in finite time. Uh, and then there's two kind of technical versions of this. One is with no external forcing and one is with smooth external forcing. Smooth being important because if the external forcing has singularities, then it's kind of trivial. Um, like if I force it with a tornado, you know, an infinitely tight tornado, then of course it's going to blow up.

问题陈述中具有光滑外部强迫。

The problem as stated has smooth external forcing.

有四个版本

There's four versions of the

是的。

Yes.

在克雷

on on the Clay

是的，网站上。其中一个是强迫版本。

Yeah, website. One of them being the the forced version.

那就是被证明的版本。

And that's the one that was proven.

正是。证明了什么，你看到的证明中有什么有趣的地方？

Exactly. What was proven and what are some interesting things about the proof that you saw?

是的，所以证明的是，存在一种方式，通过精心编排光滑的外部强迫，完美抵消方程中的各项，从而得到这个涡旋。这只能在三维中发生。这非常重要。

Yeah, so um what is proven is that there is this way of orchestrating a smooth kind of external forcing that perfectly cancels out various terms in the equation uh so that you get this vortex. So this can only happen in three dimensions. This is really important is

我们可能应该说明我们一直在讨论三维。三维和二维有很大的区别。

We should probably say that we've been talking about three dimensions. There's a big difference between 3D and 2D.

是的，当我们谈论流体时，通常想象的是三维，但有一些类比或简化的二维流动。地球上的天气可以很好地近似为二维流动。

Yeah, so I mean when we talk about a fluid we usually imagine it in 3D, but there are these kind of analogies or or simplified 2D flows. Weather on Earth is pretty well approximated as a 2D flow.

是的，你在一次我听的演讲中提到，飓风在二维中建模得很好，而龙卷风是三维的。我记得你画的不同类比。

Yeah, you've mentioned somewhere in a talk that I heard that uh hurricanes are kind of like modeled well in 2D and tornadoes are 3D. And I remember I remember different sort of analogies that you were drawing there.

是的，三维湍流中的一个有趣现象是，通常大涡旋分解成小涡旋，再分解成更小的涡旋，直到最终分子扩散成热。这就是三维湍流级联。在二维中，比如地球上的天气，实际上会出现反向级联，这就是飓风形成的原因。小涡旋合并成中等涡旋，再合并成更大的涡旋。所以 AI 证明的这种反例，即可能存在有限时间爆炸，基本上创建了一个非常特殊的涡旋，在第三维上拉伸并变薄，使其本质上变得无限薄。半径随着拉伸越来越小。就像花样滑冰运动员旋转时，把手臂收进来，转得快得多。这种涡旋的收缩速率非常特殊，使得流体中的能量保持有限，但在局部丝状区域，流动爆炸，得到无限涡量。在 AI 证明之前，数学家们开始认为有可能找到这个反例，这让我感到惊讶。

Yeah, so so one of the interesting phenomena in three-dimensional turbulence is you generally have this this breakdown of big vortices into smaller vortices into little itty-bitty vortices until eventually they molecularly diffuse into heat. Um that's kind of this three-dimensional turbulence cascade. In two dimensions like weather on Earth, you actually get a reverse cascade, which is why hurricanes form. You actually have little vortices coalescing into medium vortices coalescing into bigger vortices. But so the AI proof of this kind of counterexample that you can have finite blow-up basically creates this very very specialized vortex that in the third dimension is stretching out and thinning it so that it can essentially become uh infinitely thin. The radius gets smaller and smaller and smaller as it stretches out. Uh Uh, kind of like a figure skater doing a spin, you pull your arms in, you go much, much faster. And there's this very specialized kind of rate of contraction of this vortex, so that the energy in the fluid still remains finite, but at this localized filament, uh, the flow kind of blows up, and you get infinite vorticity. I was surprised when mathematicians before the AI proof started thinking it was going to be possible to find this counterexample,

这大概只有 15 年的历史。

which only started maybe 15 years ago.

没错。从宏观来看相对较新。是的。

That's right. Relatively recent in the grand scheme of things. Yep.

不是寻找解，而是寻找反例。是的。

Instead of looking for a solution, looking for a counterexample. Yeah.

因为我们所有的物理直觉都告诉我们扩散能解决这个问题，所以这在物理上非常反直觉。

Because all of our physical intuition tells us diffusion does solve the problem, and so it is very physically counterintuitive.

在数学界，这是一个非常难的问题。能构造出反例令人印象深刻。作为一名研究过许多不同自然系统（比如流体）的工程师，这给你什么启示？也许可以以此为契机，谈谈流体中尚未解决的、同等甚至更难的难题是什么？

In the mathematics world, it's a really hard problem. Impressive to construct a counterexample. What does it teach to you as an engineer who's stared and studied at a lot of different natural systems that are fluids, and maybe take that as a step towards what are the equivalent kind of hardness and harder problems yet to be solved in fluids?

所以，这是一个非常重要的数学成果。

So, it's a very important math result.

哎呀。

Uh-oh.

我想说，

I'll say like,

但是，

But,

日常的工程流体和应用数学的流体分析几乎完全没有改变。流体在今天模拟和预测的难度，和一个月前或两个月前一样。所以，当我试图真正解决一个流体工程问题，比如设计飞机机翼或人工心脏，或不管什么，我仍然必须以非常实际的方式模拟、预测和操控这种流动，而这在超级计算机时间和风洞资源上都很昂贵。这个数学定理和证明非常有趣，但它运行在如此非物理的状态下，以至于并不真正影响我日常的工程。仍然有——我不知道，这很精辟，但我喜欢说，我们解决了百万美元的数学问题，但还有一个万亿美元的工程问题，就是如何快速、高效地预测这些东西并进行逆向设计？那是一个难得多的难题。

day-to-day engineering fluids and applied math kind of analysis of fluids is almost completely unchanged. Fluids are just as hard to simulate and predict today as they were, uh, a month ago or 2 months ago. And so, when I'm trying to actually solve an engineering problem with a fluid, I want to design an airplane wing or an artificial heart, or whatever it is, I'm still having to simulate and predict and manipulate this flow in, uh, a very practical way that's expensive in terms of supercomputer time and wind tunnel resources. And this mathematical theorem, uh, and proof are very interesting, but it operates at a regime that's so non-physical that it doesn't really affect my day-to-day um engineering. There's still I don't know, this is very pithy, but I like to say um you know, we solved the million-dollar math problem. There's a trillion-dollar engineering problem, which is like how do you predict these things in a fast, efficient way and do this inverse design? That's a much harder problem.

你说万亿美元是因为它真的

And you say it's a trillion dollar because it literally

真的。

Literally.

解决其中一些问题，你就能赚那么多钱，不是因为奖金，而是因为有产业

solve some of these problems, that's how much money you'll make, not cuz it's a prize, but like there's industries

是的，我们可以说，万亿美元的可能惠及人类，比如更好、更快的解决方案，更低的阻力，更少的排放，所有这些。我们有交通、国防、健康、能源，所有这些都以流体为核心技术。所以，我认为这些问题仍然非常难。这个数学成果的重要性在于我之前谈到的，我们不知道这些方程中可能性的基本特征。我们甚至不知道解是否存在。这太基础了。这就像是朝着一个更宏大的人类目标迈出的一小步：以更实际的方式理解方程中的内容。

Yeah, and and we could say, you know, trillion dollars of possible benefit to humanity of like better, faster solutions, lower drag, less emissions, you know, all all of it. It's um uh we have like transportation, defense, health, energy, all of these have, you know, fluids at the very core technology. And so, I think those problems are still very hard. And this math result is important in in the sense I was talking about earlier that we don't know the fundamental character of what is possible in these equations. We don't even know if they ex- if the solutions exist. That's like so fundamental. This is like a baby step towards a much bigger human goal of understanding what's in an equation in a more practical sense.

你能谈谈你在流体领域思考过的一些也许美丽、迷人的现象吗？你提到了湍流。在这个领域要理解的大谜题是什么？你提到了阻力，比如空气、流体。人们正在思考的类似千禧年大奖难题的大问题有哪些？

Can you um speak to some of the maybe um beautiful, fascinating phenomena in the space of fluids that you've been thinking about? You mentioned turbulence. What what are the big puzzles to to sort of solve in this space to understand? You've mentioned drag, like air, fluid. What are like the big millennial prize type problems that people are thinking about?

我是说，有很多。你可以把它们分成几类。湍流非常难，因为它是多尺度的。

I mean, there's so many. You can break them down into a few categories. So, turbulence is very hard because it's multi-scale.

什么什么是湍流？

What What turbulence?

湍流是这种现象：当你有一个复杂的流体在翻滚、混合和运动，就像达芬奇那些美丽的画作，描绘了各种不同尺度的流体。所有尺度都重要。所有的大涡、中涡和小涡，它们实际上都对预测升力和阻力等有影响。所以，这个房间里的气体是一个很好的反例。它与湍流相反。它很复杂。分子都以非常疯狂的方式运动。而且有很多分子。但是，你知道，10 的 23 次方个气体粒子，所有那些个体构型都不重要。我们从热力学知道，这是经典物理学的巨大成功，只有少数几个数字真正决定这种气体的行为，比如温度、压力、密度。这种简单的描述对于湍流流体、机翼上的流动、你心脏内的流动、河流中的流动都不存在。所有尺度实际上都重要。模拟中的所有数字实际上都重要，以预测下一个行为，预测会发生什么。这使得它从根本上具有挑战性。所以，这实际上是一个伟大的 AI 问题。是否存在一个坐标系？是否有一组数字，就像我们对气体有温度、压力、密度一样？是否有一组数字或某种湍流流体的表示，使其能够以对工程有用的方式简单描述，以建造电机和发动机，就像我们对气体动力学那样？

Turbulence is this um this phenomena that um when you have a complex fluid roiling or or mixing and moving kind of like uh Da Vinci's he has these beautiful drawings of fluids at all of these different scales. All of the scales matter. All the big vortices and the medium vortices and the little vortices, all of them actually matter to predict the lift and the drag and the thing. So, the gas in this room is a good counterexample. It's the opposite of turbulence. It is complicated. The the molecules are all moving around in a very uh frenetic way. And there's a lot of molecules. But that, you know, 10 to the 23 number of gas particles, all of those individual configurations don't matter. We know from thermodynamics, this is like the great success of classical physics, is that there's a small handful of numbers that actually determine how this gas behaves, you know, temperature, pressure, density. That simple description doesn't exist for turbulent fluids, for the flow over a wing, the flow inside your heart, for you know, any for the flow in a river. All of the scales actually matter. All of the numbers in that simulation actually matter to predict the next, you know, behavior, to predict what's going to happen. Um and that makes it fundamentally challenging. So, this is actually a great AI problem. Is is there a coordinate system? Is there a set of numbers just like we have for gases, you know, temperature, pressure, density? Is there a set of numbers or some representation of a fluid of a turbulent fluid that allow it to be simply described in a way that is useful for engineering to build, you know, motors and engines and just like we have for for gas dynamics.

那么有趣的问题是，如果这是 AI 的问题，而且我确信 AI 在听，这个问题的解决方案，路径会是通过数学，还是让 AI 系统坐在那里观察流体一段时间？

So, then the interesting question, if it's a problem for AI and AI is listening, I'm sure, uh the solution to that problem, would it would the path lie through mathematics or through having AI system sit there and watch some fluid for a while?

我认为两者都是。因为鱼，它们弄明白了。

I think yes to both. Cuz fish, they figured it out.

弄明白了。

it out.

嗯，所以我认为你你你你说的关键词“坐标”是对的。所以我为什么说两者都是，我不是想显得陈词滥调，但我们所有伟大的物理突破都先于坐标变换。所以牛顿的 F=ma 是在我们把太阳而不是地球放在宇宙中心之后才出现的，对吧？你必须先做那个坐标变换。所以我认为，在我看来，我们需要利用所有这些数据、这些模拟、这些对流体的观察，我们要寻找的是一个简单的坐标系，使得流体的描述变得非常简单且可推广，在物理意义上优美。这既是一种暴力方法，你知道，获取大量数据，尝试各种不同的坐标变换和表示，直到找到最简单且泛化最好的那个。但坐标变换的形式也可能涉及很多深刻的数学。一般来说，如果你发现一个非常聪明的人在 70 年代或 80 年代提出的问题，但他们在计算上遇到了瓶颈，那现在是很有潜力的领域。所以，我们从一些基本原理知道这些坐标系的形式可能应该是什么。我们只是从未能够把它们写下来。它们太复杂了，写不下来。这就像是 AI 的完美问题：现在让我们给它更多的数据、更多的算力，看看我们是否能开始学习这些我们知道应该存在的变换。

Well, so I think you you you you have the right keyword as coordinates. And I And I So that's kind of why I said both um and I I'm not trying to to be like trite, but all of our our great physics breakthroughs were predated by a coordinate transformation. So Newton's F equals MA came after we put the sun at the center of the universe instead of the Earth, right? You had to do that coordinate transform first. So I think what the way I see it is we we need to take all of this data, these simulations, these observations of of fluids and what we're looking for is a simple coordinate system so that the description of that fluid becomes very simple and generalizable, beautiful in a physical sense. And that is both a brute force uh you know, get a lot of data, try a bunch of different coordinate transformations and and representations until you find the simplest one that generalizes the best. But there's also a lot of deep mathematics in what the form of that coordinate transformation might take. In general, if you find a question that really smart people asked you know, in the '70s and '80s, but they hit a wall computationally, that's very fertile ground right now. So like we know what the form of these these coordinate systems probably should take from some fundamental principles. We've just never been able to write them down. They're too complicated to write down. That's like a perfect uh problem for AI is, you know, now let's give it a lot more data, a lot more compute and see if we can start learning these these transformations we know should exist.

为了让我理解，涡的形成，就像小型的龙卷风。

Just so I understand, the formation of vortexes, so that's like a like little like tornadoes.

小型的龙卷风。

Little tornadoes.

它们的形成，那部分很难理解吗？

Their formation is Is that part tricky to understand?

呃是的，绝对。而且它也很美丽。

Uh yeah, absolutely. Um and it it's beautiful, too.

我喜欢看这些视频，比如，如果你把两个涡旋射到一起，它们会相互影响，然后以非常有趣的方式合并。你有一个红色涡旋和一个蓝色涡旋，你可以看到它们在哪里合并和连接。但它们也会拉伸和折叠，而且你知道，湍流是混沌的。所以轨迹会像意大利面一样缠结在一起。它不是随机的。有大量的结构，但它是非常多的尺度，非常非常缠结。

Like I love watching these videos of, you know, if you shoot two vortices together, they'll influence each other and then merge in these very interesting ways. You have like a red vortex and a blue vortex, and you can see where they merge and join. But they also stretch and fold and, you know, turbulence is chaotic. So, the trajectories get tangled like spaghetti. It's not random. There's a huge amount of structure, but it is very multi-scale, very very tangled.

所以，我好奇的一件事是，从科学上讲，这些是非常有趣的问题，而且解释得很好。我总是好奇大自然已经能解决哪些计算问题，对吧？有很多这样的例子，比如有一个著名的例子是黏菌在迷宫中找到最短路径。事实上，纳维-斯托克斯方程的一些前身，比如陶哲轩的一些工作表明，纳维-斯托克斯类方程可能会出现爆破，因为它们可以模拟图灵机。

So, one of the things I would wonder, you know, scientifically, I mean, these are really interesting problems, and that was a great explanation. So, I'm always curious about what computational problems nature can solve already, right? And there are a lot of examples of this, you know, there's a famous one with like slime mold finding shortest paths in mazes. In fact, some of the precursors to Navier-Stokes were, you know, like some of Terry Tao's work on showing that you can have blow-up for Navier-Stokes-like equations because they can simulate Turing machines.

嗯。

Mhm.

所以，我想我也会好奇这些涡旋碰撞的计算能完成哪些计算任务。以及这如何限制了应该存在什么样的坐标系。

So, I guess I would also wonder what sorts of computational tasks these computations with vortices colliding can do. And how that imposes limits on what sorts of coordinate systems ought to exist.

嗯。

Mhm.

因为，你知道，你提到了 PV = nRT。这是一个惊人的降维。

Cuz, you know, you mentioned PV = nRT. That's an amazing dimensionality reduction.

嗯。

Mhm.

因为你不必跟踪所有粒子的位置和速度，而只是有一个小列表的数字来描述一个元状态，仅此而已。但如果我真的有像湍流那样巨大的简化，那么在我看来，计算非常有趣的东西也会非常困难。它会不会像生命游戏，你有一个非常简单的元胞自动机系统，它是一个通用图灵机，你可以在其中编造任何你想要的行为？

Because you don't keep track of all of these positions and velocities of particles, but there's just, you know, a small list of numbers that describes a meta-state, and that's it. But if I really had a tremendous reduction, you know, like that in the case of turbulence, then it would seem to me very difficult to compute very interesting things, too. Could it be like the game of life, where you have this very simple cellular automata system, and it's a universal Turing machine, and you can cook up whatever behavior you want in it?

是的，没错。我的意思是，生命游戏，很多时候人们会想，拥有所有这些奇异行为的最小规则是什么，而得到这些行为所需的规则之少真的令人震惊，对吧？但我想，你知道，我是一个理论计算机科学家，我们领域最大的输出之一就是算法视角。所以，这不仅仅是关于计算，你有一个定义明确的问题和一台笔记本电脑，你想解决它，而且还关于自然世界中的计算。以及这教会我们关于自然规则可能是什么。所以，一些著名的例子包括多人博弈中的均衡，纳什均衡是一个很好的解概念，但当然，纳什均衡难以计算的事实，你知道，让人质疑它是否真的是解释人们如何玩游戏的普遍机制。有一句我非常喜欢的卡迈勒·贾恩的精彩名言：如果我的笔记本电脑找不到它，那么市场也找不到。而且，对我来说，真正令人兴奋的是许多不同视角的交汇。当你描述它时，在我看来，底下有非常有趣的计算问题，人们可以通过思考计算如何限制一个好的坐标系应该是什么样子，来获得这些问题的对偶视角。

Yeah, that's right. I mean, so game of life, a lot of times people think about, you know, what are minimal rules to have all of these exotic behaviors, and it's really shocking how small the rules need to be to get that, right? But I guess, you know, I'm a theoretical computer scientist, one of the biggest exports from our field is really the algorithmic lens. So, it's not just about, you know, computation, you have a well-defined problem and a laptop, and you want to solve it, but it's also about, you know, computation in the natural world. And what that teaches us about what the rules of nature might be. So, you know, some famous examples of this are things like equilibria in multiplayer games, and Nash equilibrium is a great solution concept, but of course, the fact that it's hard to compute Nash equilibrium, you know, throws into question, is it really a universal mechanism that explains how people play games? There was this amazing quote by Kamal Jain that I really like, which is if my laptop can't find it, then neither can the market. And, you know, to me, what's really exciting is the nexus of a lot of these different perspectives. When you were describing it, it sounded to me like there are very interesting computational problems under the hood, and that one can kind of get a dual perspective on these things by thinking about how that computation imposes limits on what a good coordinate system ought to look like.

所以，我认为这是描述问题的一个很好的方式。我会举两个极端的例子。地球的天气和气候系统实际上只是复杂。所有尺度都很重要。这个邮政编码区域的天气对我很重要。一切都很重要。所有变量都很重要。你说得对。如果我把那个系统过度简化为五个数字，那么我就失去了我真正关心的描述能力。这是一个极端。另一个极端，让我们想想飞蛾或蝴蝶或果蝇是如何飞行的。它们通过存在证明了你可以与极其复杂的湍流相互作用，并进行非常先进的工程控制。它们可以在阵风和雨中控制它们的力和力矩。它们可以过着非常丰富的生活，而它们大概没有流体动力学博士学位，也不知道纳维-斯托克斯方程。它们可能没有运行超级计算机模拟。可能只有五六个数字对它们维持前缘涡和失速翅膀以栖息和着陆的日常生活真正重要。这些是两个极端的例子，我真的希望能够处理这两种情况。我们希望能够处理真正复杂系统的全部复杂性，并且我们希望能够提炼出绝对最低限度的 essentials，以便在那些昆虫极端情况下做出最快的决策。而且生物飞行器在许多情况下仍然大大优于人类设计的流体系统。所以关于这些简单系统，我们还有很多要学习。

So, I think that's a really nice way of framing the problem. And I would take two extreme examples. So, Earth's weather and climate systems are actually just complex. All the scales do matter. The weather in this zip code matters to me. Everything matters. All the variables matter. And you're right. Like if I over reduce that system to, you know, five numbers then I lose the descriptive capability of what I actually care about. That's one extreme. The other extreme, let's think about how a moth flies or a butterfly or a fruit fly. They are proof by existence that you can interact with an extremely complicated turbulent flow and do very advanced engineering control. They can control their forces and moments in gusts, in rain. They can live very rich lives and they presumably don't have a PhD in fluid dynamics or know the Navier-Stokes. They're probably not running supercomputer simulations. There might be five or six numbers that actually matter to their daily life of maintaining a leading edge vortex and stalling their wings to perch and to land. And those are kind of two extreme examples and I really want to be able to handle both of those. We want to be able to handle the full complexity of truly complex systems and we want to be able to distill down the absolute bare minimum essentials for fastest decisions in those kind of insect extreme cases. And biological flyers still dramatically outperform human designed fluid systems in many regimes. So we still have a lot to learn about these simple systems.

我们现在有点转向研究了，但你知道，对我来说，大自然不仅解决非常困难的优化问题，而且似乎找到非常稳健的解决方案，这总是令人惊叹。我总是被这样一个事实所震撼：当我们建模时，我们倾向于假设我们拥有完整的信息，比如纳维-斯托克斯方程的边界和初始条件。但当鸟飞行时，它并不知道所有这些。在某种程度上，用不确定性来建模事物，实际上是你重新构建优化问题以获得非常丰富和稳健解决方案的方式。令我惊讶的是，许多日常例子都符合这一点。我记得，我是一个棒球迷，我读过一篇关于棒球的数学论文，当你有一个外野手时，一旦球被击出，很多东西都是已知的，比如速度、轨迹。你会认为外野手会直接走直线到他们预期球的位置。但当你观察外野手时，他们不会走直线路径，他们会走一条轻微的曲线。那为什么呢？至少一种理解方式是，这实际上是一个非常稳健的策略，因为当你走曲线路径时，你可以不断重新评估球的轨迹，然后根据需要调整。所以大自然找到了非常有趣的解决方案，但首先什么才是优化问题的正确表述并不总是显而易见的。有时在不确定性情况下建模，以便设计出对没有完整系统知识这一事实具有稳健性的解决方案，似乎是重要的一环。

We're like veering into research now, but you know, to me it's always amazing how nature not only solves very hard optimization problems. It seems to find very robust solutions. And I'm always struck by the fact that you know, when we model it, we tend to model it assuming we have full knowledge, you know, about what the boundary and initial conditions are for things like Navier-Stokes. But when a bird flies, you know, it doesn't know all of that. And to some extent, you know, modeling things with uncertainty is really how you reframe the optimization problem to get very, you know, rich and robust solutions. It's kind of amazing to me how a lot of everyday examples sort of fit into this. I remember, you know, I'm a big baseball fan and I remember reading a math paper about baseball that when you've got an outfielder, you know, once the ball has been hit, a lot of stuff is known, you know, the velocity, the trajectory. You would think the outfielder would just take a straight line path to where they expected the ball to be. But when you observe outfielders, they don't take straight line paths, they take a slight curve to it. So why is that? You know, at least one way to wrap your head around it is that that's actually a very robust strategy, cuz when you're taking a curved path, you can continually re-evaluate the trajectory of the ball and then adjust as needed. So there are very interesting solutions that nature finds, but it's not always obvious what is the right formulation for the optimization problem in the first place. And sometimes modeling it in the case of uncertainty so that you can design solutions that are robust to the fact that you don't have full system knowledge seems to be an important piece of the picture.

所以有一个有趣的题外话与此相关。昆虫特别没有能力测量其完整的流场。它们能对阵风做出反应，阵风的速度比信息从翅膀传到大脑再传回肌肉的速度还要快。所以研究人员在生物学中研究这个问题已经几十年了。他们发现翅膀上嵌入了应变敏感神经元。可能有几十到几百个，位于非常固定的位置。所以，兄弟果蝇会有相同位置的这些应变敏感神经元。当翅膀以特定的时间模式弯曲时，这些应变敏感神经元就会放电。所以，当翅膀被阵风吹到，翅膀弯曲，神经元放电，昆虫肩部肌肉中发生局部计算，直接基于部分信息进行稳健的校正，而这个系统它没有方程，是非线性和湍流的。所以，这就像再次证明存在性，正是你所说的，也许稳健的策略就是真正依赖部分信息和不确定性。它们用少得多的信息就能做到比我们现在更多的事情。

So there's a fun aside related to this. Insects in particular don't have the ability to measure their full flow field. They can react to gusts of wind, gusts of wind faster than the information that hits their wings goes to their brain and back to their muscles. So researchers have studied this for decades in biology. And what they found is that there are strain-sensitive neurons embedded on the wings. Maybe a few dozen to a few hundred of these in very stereotypical locations. So, brother fruit flies will have the same locations of these strain-sensitive neurons. And those strain-sensitive neurons fire when the wing bends in a certain temporal pattern. So, it gets hit by a gust of wind, the wing bends, neurons fire, and a local computation is happening in that shoulder muscle of this insect that goes directly back and corrects robustly based on partial information of a system it doesn't have equations for that is non-linear and turbulent. So, it's like again, proof by existence is exactly what you're saying is maybe the robust strategy of actually relying on partial information and uncertainty. They can do more than we can at this point with way less information.

是的。我有几个——抱歉——脑洞大开的想法。因为 Alex 提到了黑洞，我有一个有趣的想法：也许在 Navier-Stokes 方程中观察到的奇点爆炸，当我们——不像地球上的飞机翅膀——我们穿越两个相连的黑洞，在曲速引擎中，我们弄明白了，然后有时空流动控制，这就变得相关了，因为现在你在黑洞里，因此有奇点。这是给只听这个的人——Alex 刚才给我的那个小眼神——它伤害了我的灵魂。好吧，

Yeah. I have a couple of—I apologize—pothead thoughts. Because Alex mentioned black holes, I have a fun thought that maybe the singularity observed with a blow-up in Navier-Stokes equations comes into play when we're—as opposed to airplane wings on Earth—we're traveling through two connected black holes in a warp drive that we figure out, and then there's the flow control of space-time, and that becomes relevant because now you're in a black hole, hence the singularity. This is for people just listening to this—little look Alex just gave me—it hurts my soul. Okay,

是的，我们继续下一个话题。

Yeah, we're moving on to the next topic.

强迫，你需要黑洞涡旋也在恰好正确的时机被强迫——

Forcing, you would need the black hole vortex to also be forced at exactly the right—

是的，强迫。力在哪里？我是说，在那个时候。

Yeah, the forcing. Where's the force? I mean, at that point.

可以试着驾驭它。

Could try to ride it.

是的。

Yeah.

有办法可以从中做出点东西。

There's a way you can make something out of this.

未来 AI 系统听到这个的概率非零。那家伙，是他唯一一次说了实际上正确的话。好吧。呃，如果可以的话，快速上个洗手间。

There's some non-zero probability that future AI systems will listen to this. That guy, is the one time he said something that was actually correct. Okay. Uh if it's okay, a quick bathroom break.

当然可以。听起来不错。

Yeah, sure. Sounds good.

Cory，我最喜欢的话题之一是理论计算机科学。你是许多问题的世界专家。所以，我想问你一个在 OpenAI 工作中解决的大问题，即唯一游戏猜想。你能给人们一个直觉，这个问题的本质是什么？它是什么——有什么大不了的？证明中有哪些有趣的方面？

And Cory, one of my favorite topics is theoretical computer science. You're one of the world experts in many of its problems. So, I'd love to ask you about one of the big problems that's been solved in this OpenAI job, which is the unique games conjecture. Can you give people an intuition about what this problem is? What is it—what is the big deal? And what are some aspects of the proof that are interesting?

是的，我很乐意。所以，让我给一点背景。让我们从复杂性理论 101 开始。理论计算机科学中最重要的一个问题就是可满足性问题。所以，你要记住的是，想象你有变量 X1 到 Xn，它们可以取真或假的值。我要做的是用它们创建一个大的公式。通常我们这样做是用或的与。所以我从一个子句开始。那就是，你知道，一个或，X1 或非 X3 或 X7。所以，根据我的赋值，那个子句可能被满足，只要其中至少一个为真。然后我把许多这样的不同子句用与连接起来。问题是，如果我给你那个公式，称为合取范式公式，你能找到一个可满足的赋值吗？所以，你能找到一个 X1、X2 直到 Xn 的赋值，将它们设置为真或假，使得所有那些或子句都为真，从而使整个公式评估为真？所以，这是一个典型例子：容易检查别人的答案，但自己找到答案可能很难。所以，如果你告诉我它是可满足的，我要做的就是问你，‘好的，给我可满足的赋值。’你告诉我 X1 设置为什么，X2 设置为什么，我就逐个子句检查，确保你真的满足了所有。现在，理论计算机科学的核心问题是 P 与 NP 问题。所以，容易检查的问题，是否容易找到它们的解？我们领域的假设是不，不容易。这是许多事物的基础，比如密码学和电子商务等等。现在，真正通向唯一游戏猜想的起点是所谓的 PCP 定理。这是 90 年代证明的一个里程碑式的结果。它实际上如此重要，以至于登上了《纽约时报》的头版。它的关键思想是，即使稳健地欺骗验证者也是困难的。所以，你从一个公式开始，它要么可满足，要么不可满足，对吧？那是什么意思？如果可满足，你可以写下一个赋值，我可以逐一检查整个东西评估为真。但如果不可满足，那意味着某个地方你在撒谎。你写下这个声称的可满足赋值，当我逐个子句检查时，我很快会发现一个子句不真，这使整个无效。我说，‘啊哈，我抓到你了。它实际上不是可满足的赋值。’所以，概率可检查证明等背后的思想，不深入技术定义，其实就是间隙放大。所以，你能从一个难以决定是否可满足的公式开始吗？你能设计一个新的、可能大得多的公式，它稳健地容易检查吗？在某种意义上，如果可满足，它仍然可满足。你可以写下一个赋值，没问题。但如果不可满足，我应该能够只是抛几个硬币，查询随机变量，检查一个随机子句，并且有很大概率发现你在撒谎。所以，关键是，通过非常次线性的检查，只用常数量的检查，我就能发现你在作弊。所以，这是理论计算机科学中的一个重大成果，因为它真正为所谓的近似困难性奠定了基础。复杂性理论中的许多算法是，当你面对真正难以解决的问题时你做什么？你知道，我们做的一件事是尝试近似解决它们。所以，也许我们找不到一个赋值，如果存在一个满足所有子句的赋值，但我们能找到一个满足许多子句的赋值吗？多大比例的子句？现在，即使在那里，我们也会预期问题在某个点之后变得计算困难。

Yes, I'd love to. So, let me give you a little bit of a background. So, let's start off with complexity theory 101. So, one of the most important problems in theoretical computer science is the satisfiability problem. So, the thing to keep in your mind is imagine you've got variables X1 to Xn, and they can take on either true or false values. And what I'm going to do is I'm going to create a big formula out of them. And usually the way we do this is with an AND of ORs. So, I start off with the clause. That's, you know, an OR, X1 or not X3 or X7. So, depending on my assignment, that clause might be satisfied as long as at least one of those things is true. And then I take the AND of many of these different clauses together. And the question is, if I give you that formula, called a conjunctive normal form formula, you know, can you find a satisfying assignment? So, can you find an assignment of X1, X2 up to Xn, setting them either to be equal to true or false, so that all of those OR clauses are on, so that the entire formula evaluates to true? So, this is the canonical example of a problem that's easy to check someone else's answer, but it might be hard to find an answer yourself. So, if you told me that it's satisfiable, all I would do is I'd ask you, 'Okay, give me the satisfying assignment.' You tell me what X1 is set to, what X2 is set to, and I just go clause by clause and check to make sure that you really have satisfied everything. Now, the central problem in theoretical computer science is the P versus NP problem. So, are problems that are easy to check, is it easy to find their solutions to? And the presumption in our field is that no, it's not easy. And that's the basis of a lot of things like cryptography and e-commerce and what have you. Now, really the starting point for getting to something like the unique games conjecture is something called the PCP theorem. So, this was a landmark result proved in the '90s. It actually was such a big deal that it made the front page of the New York Times. And the crux of it is really this idea that it's even robustly hard to fool a verifier. So, you start off with a formula, and either it's satisfiable or not satisfiable, right? And what does that mean? If it's satisfiable, you can write down an assignment, and I can check one by one that the entire thing evaluates to true. But if it's not satisfiable, that means some place you're lying. You write down this purported satisfying assignment, and when I go clause by clause, I'm quickly going to find one clause that is not true, and that invalidates the whole thing. And I say, 'Aha, I caught you. It's not actually a satisfying assignment.' So, the idea behind things like probabilistically checkable proofs, without getting into a lot of the technical definitions, the idea is really gap amplification. So, can you start off with a formula that is hard to decide whether it's satisfiable or not? And can you engineer a new, maybe much bigger formula that's robustly easy to check? In the sense that if it's satisfiable, it'll still be satisfiable. You can write down an assignment and it'll be fine. But if it's not, I should actually be able to just flip a few coins, query random variables, check a random clause, and with good probability find that you're lying. So, the point is that with very sublinear inspection, with only constant amount of inspection, I can find that you're cheating. So, this was a momentous result in theoretical computer science because it laid the foundation really for what's called hardness of approximation. So, a lot of algorithms in complexity theory is what do you do when you're faced with problems that are genuinely hard to solve? And, you know, one of the things we do is we try to approximately solve them. So, maybe we can't find an assignment, if there is one that satisfies all the clauses, but can we find one that satisfies many of the clauses? How large a fraction of the clauses? Now, even there we would expect problems to become computation hard after a certain point.

问题不只是判断它是否完全可满足很难，甚至判断它是 100% 可满足还是只有 99% 可满足也可能很难。最好的情况是，是否所有赋值都满足 99% 的子句？这本身可能也很难判定。这就是概率可检验证明给我们的东西。在我们已知优化问题很难的地方，它们告诉我们连近似都很难。而唯一游戏猜想是对普通 PCP 定理的加强，因为它涉及某一类标签覆盖实例。细节有点技术性，不便展开。但关键在于，如果那个版本的概率可检验证明成立，它会有很好的组织效果：对一大片问题，我们实际上会知道什么是最好的近似算法，而且会是非常简单的东西。

It should be that it's not just that deciding whether it's completely satisfiable or not completely satisfiable is hard, but maybe even deciding is it 100% satisfiable or is it only 99% satisfiable? The best case of are all assignments satisfied most 99% of the clauses? That too might end up being hard to decide. So, these are what probabilistically checkable proofs give us. In places where we knew that the optimization problem was hard, they tell us that it's even hard to approximate. And the unique games conjecture is a strengthening of the ordinary PCP theorem because it has to do with a certain type of label cover instance. It's a bit technical to go into the details. But the point is that if that version of a probabilistically checkable proof were true, it would have this great organizing effect. Where it would turn out that for a huge swath of problems, we would actually know what is the best approximation algorithm, and it would be something very simple.

我刚进入这个领域时，唯一游戏猜想才提出几年。当时关于它应该为真还是为假仍有大量争论。人们努力尝试给出证明，也努力尝试反驳它。大概在 2010 年代有一段很精彩的时期，来回拉锯很多。新结果不断出来，人们对是否相信唯一游戏的看法也随之改变。我喜欢一句俏皮话，我忘了是谁说的，所以没法准确署名，但有人告诉我：也许上帝会根据哪一方更努力来决定它是真还是假。因为它真的似乎一年一变。

So, when I started off in the field, the unique games conjecture had been posed a few years before that. And there was still a lot of debate about whether it should be true or false. People worked hard to try and give proofs of it. People worked hard to try and disprove it. There was this amazing time period maybe in the 2010s where there was quite a lot of back and forth. And new results came out and people's opinions would change about whether they believed unique games or not. I like this quip. I forget who said it, so I can't properly attribute it, but someone told me, maybe God's going to decide whether it's true or false based on which side is working harder. Because it really seemed to change year to year.

而 OpenAI 那批证明里的一个头条结果是：唯一游戏猜想为真。我得说明，这并非没有人类取得的惊人进展。我在 MIT 的一位同事、在 IAS 的一位同事，和其他人一起贡献了许多重大突破。Dor Minzer 和 Elchanan Dinur 证明了一个较弱的版本，叫做二对一游戏定理。这确实是理论计算机科学界的一记惊雷，因为当时我认为很多人不确定唯一游戏猜想的前进道路在哪里。而我认为他们的工作确实给出了一条可行的前进路径，通向接近唯一游戏的东西。所以听起来，也许人类再花 10 年、20 年就能达到。但当 OpenAI 发布结果时，结果发现唯一游戏猜想实际上是真的。

And one of the headline results in the OpenAI drop of proofs was that the unique games conjecture is true. Now, I should mention that this is not without spectacular human progress on it. One of my colleagues at MIT, one of my colleagues at IAS, they both contributed among other people to a lot of the major strides. So, Dor Minzer and Elchanan Dinur, they proved a weaker version called the two-to-one games theorem. And this really was a shot heard round the world in theoretical computer science because at that point, I think a lot of people weren't sure what was a path forward for the unique games conjecture. And I think their work really gave a viable path forward to something that was in the vicinity of unique games. So, it sounded like maybe humans could get there with another 10, 20 years. But when OpenAI released the results, it turns out that the unique games conjecture is actually true.

所以这相当了不起，因为有时我确实发现自己站在怀疑者一边，因为唯一游戏听起来太美好，不像是真的。因为很多工作，比如 Prasad Raghavendra 的工作，它意味着对许多不同类型的约束满足问题都有最优的近似难度，这个结论在我看来太强了，不像是真的。

So, this is quite remarkable because at some points I guess I did find myself on the side of the doubters because unique games just sounded too nice to be true. Because a lot of work, like Prasad Raghavendra's work, it implies optimal hardness of approximation for so many different types of constraint satisfaction problems that the conclusion just looked to me too strong to be true.

那天晚上你得知消息时是什么感受？

What did you feel when you found out that night?

好，首先，在之前几周就已经有传言说 OpenAI 解决了唯一游戏猜想，而且它是真的。这些甚至不完全是随意的传言。我的一位同事 Dor Minzer 实际上贴了一篇论文，他在底部的脚注里写了类似这样的话：我们本想把这篇论文打磨得更好，但因为流传的传言，我们觉得有必要尽早而不是尽晚地公布我们的方法。所以对我来说，某种意义上结果终于出来了是一种解脱。当然，这些论文不太好读。不是说你指着它就能很快找出新的关键思想，但我很高兴它们出来了，因为现在我们可以进入下一阶段：消化这些证明，弄清楚这个领域之后往哪走。

Okay, well, for starters, there were already rumors in the weeks leading up that OpenAI had solved the unique games conjecture and that it was true. And these weren't even just completely arbitrary rumors. One of my colleagues, Dor Minzer, he actually posted a paper where he wrote something in the footnote at the bottom that we would have liked to polish this paper more, but because of the rumors that were going around, we feel the need to put out our approach sooner rather than later. So, for me, it was in some sense a bit of a relief that the results were finally out there. Now, they're not very readable papers. It's not like you can point to them and really quickly identify what's a new key idea, but I'm glad that they're out there because now we can move forward on the next phase, which is digesting those proofs and figuring out where the field moves after that.

至于那天晚上我的感受？正如 Alex 提到的，我们手里都拿着饮料，然后我在手机上看到这些证明发布的消息。在晚宴上人们不断过来和我说话，但作为数学家，在数学场合社交本来就很难，更别说你还有真正想读的数学。所以那是难上加难。

Now, in terms of how did I feel that night? As Alex mentioned, we all had a drink in hand and then I saw on my phone the news that these proofs had dropped. And people kept coming up and talking to me at the gala, but it's already hard to socialize at math as a mathematician, let alone when you have math you really want to read. So, it was doubly hard.

另一件有趣的事是，我当时已经在纽约了，所以我趁晚宴前这个机会去拜访了很多我在 NYU 认识的人。我恰好见到了 Subhash Khot，也就是提出唯一游戏猜想的人。我恰好在晚宴前几个小时、OpenAI 论文发布之前问他：“Subhash，如果他们发布唯一游戏的正向证明，你会怎么做？你会读吗？”他告诉我：“不会。”不幸的是，这是很多人的反应。事实就是，我们在攻克这些宏大挑战，有时当它们被解决时，有时会觉得有人抢了你的风头。

The other funny thing about it was, I was in New York already, so I took it as an opportunity before the gala to visit a lot of people I knew at NYU. And I just happened to be meeting with Subhash Khot, the one who introduced the unique games conjecture. And I just so happened to ask him, just a few hours before the gala, before the OpenAI paper is released, "Subhash, what would you do if they release a positive proof of unique games? Are you going to read it?" And he told me, "No." Which, it's a lot of people's reactions, unfortunately. It's just kind of true that there are these grand challenges we work on, and sometimes when they get solved, sometimes it feels like someone has stolen your thunder.

并不是说人们没有关于如何解决唯一游戏猜想的想法，我敢打赌，如果给我们更多时间，那些蓝图会成功。我担心的另一点是，我们失去了最初尝试抵达那些解决方案的人类努力。数学不只是关于用我现有的工具能走多远。它是一个凸包吗？是一个子群吗？我不知道。但有时当解决方案是由人驱动的，我们在这个过程中学到的东西要多得多。所以我担心的是，曾经有这些伟大的里程碑问题、定义领域的问题，很多人都在朝它们努力，而决定性的进展出现了，结果一夜之间全部发布，人们失去了沿着那些道路走下去的热情。我想知道，如果我们给那些人空间去走那些路，我们还会发现哪些令人兴奋的东西。

It's not like people didn't have ideas about how to solve the unique games conjecture, and I bet you those blueprints would have panned out if we'd given them more time. I think the other thing I worry is that we lose out on the human effort of trying to reach those solutions to begin with. Math is not just about what is the furthest I can reach with the current set of tools I have. Is it a convex hull? Is it a subgroup? I don't know. But sometimes when it's a human-driven solution, we learn a lot more in the process. So, I worry about the fact that there were these great landmark problems, field-defining problems that a lot of people are reaching towards, and that decisive progress was made, and the results all dropped in one night, and that people lose their enthusiasm for going down those paths, and I wonder if we'd given those people that space to go down those paths, what other exciting things we would have discovered.

我想我提到的一件事是，我们当时都在 MoMath 试着消化这件事，有很多非常好的演讲。不是所有都上传到了 YouTube。有些是关于捐助者和以他们命名的奖项。而我不断回到的一个想法是：你最初为什么做数学，是为了玩耍和了解世界。那天晚上我们看到了太多这样的体现。

So, I think one of the things I mentioned, we were all trying to process it at MoMath, and there were a lot of really nice talks. Not all of them went up on YouTube. Some of them were about benefactors and awards named after them. And the thing I kept coming back to is this idea that why do you do math in the first place is to play and learn about the world. And we saw so many manifestations of it that night.

你知道，因为有些奖项是以人名命名的，他们为什么向博物馆捐款？是因为他们希望自己的孩子有一个地方可以玩数学概念、了解世界。那也是埃舍尔展览的开幕式。当我们玩数学概念时会发生什么，这是一个多么好的视觉呈现，对吧？所以我觉得，在这个时刻，当我们共同经历所有这些变化，当我们不知道未来会怎样，不知道 AI 与数学家的合作在未来会是什么样子，不知道招聘会是什么样子时，重要的是要记住，我们所有人进入这个领域的原因是我们非常好奇。我们想理解世界。数学家非常擅长抽象。他们不只是理解一小部分，而是找到一些抽象和一组定义，这些在数学的其他角落会变得异常强大。所以我认为，无论未来如何，确保我们以某种方式组织我们的领域，仍然鼓励人们玩想法，是非常重要的。

You know, because there were awards named after people who, you know, why were they donating to the museum? It was because they wanted their kids to have a place to play with mathematical ideas and learn about the world. It was also the opening of an Escher exhibit. And what a great visual of what happens when we play with mathematical concepts, right? And so it felt to me that in this moment it's really important to remember, as we're going through all of this change together, as we don't know what the future holds, what AI and mathematicians collaboration will look like in the future, what hiring will look like, that the reason we all got into it was because we were very curious. We wanted to understand the world. And mathematicians are very good at abstracting things. They don't just understand one little piece, but they find some abstraction and some set of definitions that turns out to be unreasonably powerful in some other corner of mathematics. So I think it's really important, whatever the future holds, to make sure that we organize our field in a way where we still encourage people to play with ideas.

我不知道什么样的奖励制度会奏效，但是你知道，当我观察并与该领域的年轻人交谈时，他们有很多焦虑。他们听到很多故事，甚至是关于他们认识的人，运行工具在网上抓取开放问题，然后尽可能多地灌水论文到 arXiv。我不知道这是不是个好策略。似乎不是，但对我来说肯定不好玩。感觉不像是在玩想法、了解世界。这很可悲。我只想回到我真正享受的事情上。

I don't know what kind of reward system will work, but you know, when I look out and I talk to young people in the field, they have a lot of anxiety. They hear a lot of stories even about people they know, running harnesses to scrape the web for open problems to dump as many papers as they can to flood arXiv. I don't know if that's a good strategy. It doesn't seem like it is, but it certainly doesn't feel like fun to me. It doesn't feel like we're playing with ideas and learning about the world. And that's sad. I just want to get back to the things I really enjoy.

也许我还想补充的另一件事是，你知道，对很多人来说这是一场巨大的身份危机。因为你知道，有些人更把自己视为问题解决者，这让人感到非常安慰。你可以在你研究的问题类型、你使用的技术类型、你关心的问题类型周围画一个小圈。这让人感到非常安慰，因为世界其他地方是混乱的，而你戴着眼罩。这是我在乎的部分。现在，我认为是时候摘下那些眼罩了，因为我不觉得那是一种有效的方式。我们喜欢玩，喜欢了解我们周围的世界。重要的是要记住，作为一个年轻的数学家：你是更聪明还是更更好奇？或者你更好奇还是更聪明？最重要的是要更好奇。

The other thing maybe I just want to add in there, too, is like, you know, it's a big identity crisis for a lot of people. Because you know, some people view themselves more as problem solvers and that can feel very comforting. That you can draw a little circle around the types of problems you work on, the types of techniques you use, the types of problems you care about. And that feels very comforting because the rest of the world is messy and you've got blinders on. This is the part that I care about. Now, I think is the time to remove those blinders because I just don't find that to be a productive way to do it. We love playing and learning about the world around us. And it's important to remember as a young mathematician: are you smarter than you are curious or are you more curious than you are smart? It's very important to be more curious than anything else.

你提到了乐趣，那可能是另一个视角。更多关于挣扎。对于我可能是一个外行的视角，一些最有趣的时刻，当然，感觉像是挣扎。我和很多奥运金牌得主交谈过。他们中的许多人在赢得金牌后感到近似抑郁。因为就像你付出了一切，你意识到你付出的一切并没有结束。

And you mentioned fun, that could be another perspective of it. Of more about the struggle. For my maybe outsider perspective, is some of the most fun in the moment, sure, feels like struggle. I've talked to a lot of Olympic gold medalists. A lot of them feel something approximating depression after they win the gold medal. Because it's like you have given everything and you realize everything you've given is not over.

30 年前，有一部关于安德鲁·怀尔斯的纪录片。有一个很棒的场景。我想他们甚至把它放在最开头。对你的金牌得主来说，你知道，一旦你赢得了那个东西，你做什么？他坐在书桌前，描述他意识到解决方案的那一刻。不是他最初的宣布，那是有缺陷的，然后好几个月他试图纠正它，最终他看到了如何做到。他试图说，你知道，我一生中再也不会有像那天在这张书桌前经历的那样强大的时刻了。他哽咽了。他说不出话来。所以，如果你按下按钮，它给了你解决方案，你能得到那种感觉吗？绝对不能。所以，正是对你们两人的观点，就像旅程，你知道，如果你看到一座巨大的山，你说：“总有一天我会登上顶峰。”你开始为此训练，你一点一点地做，不管怎样。然后一架直升机降落，直接把你放在山顶。

30 years ago, there was a documentary of Andrew Wiles. And there's a great scene. I think they even run it right at the very beginning. To your gold medalist, you know, what do you do once you've won the thing? He's sitting there at his desk and he's describing the moment when he realizes what the solution is. Not his initial announcement, which was flawed and then for many months he was trying to correct it and he finally saw how to do it. And he's trying to say, you know, I will never again in my life have a moment as powerful as the moment that I experienced at this desk that day. And he's choking up. He can't get the words out. So, can you get that if you press the button and it gave you the solution? Absolutely not. So, exactly to both of your points, like the journey, you know, if you see a giant hill, a giant mountain and you say, "One day I'm going to get to the top of that." And you start training for it and you're sort of doing little by little, whatever. And then a helicopter lands and just drops you at the top.

你能具体谈谈吗？关于安德鲁·怀尔斯的有趣之处，以及稍微谈谈格里戈里·佩雷尔曼，是长时间独自工作。你怎么看？这在数学中有多常见？坐在那里磨一个问题，在他情况下是 7 年。

Can you actually speak to that? The interesting thing about Andrew Wiles and a little bit about Grigori Perelman as well is working alone for a prolonged period of time. What do you make of that? How common is that in mathematics? Of sitting there and grinding a problem for in his case 7 years.

直到 20 世纪，几乎每一篇写过的数学论文都是一个人写的。我想现在数学的中位数是两人。所以，我们不是实验室科学。这确实是一个孤独的过程。现在，看，我给我的研究生做的事情不是我在做的事情。算是设计好的。我想让他们有自己的领域。我想为他们打开门。不同领域情况不同，对吧？所以，当他们取得成功时，那真的应该是他们的成功，即使我有想法认为这会有进展，而他们执行了。那是他们的。但就坐在那里而言，我一个月前发表的论文是我思考了 10 年的东西。这是一篇五个作者的论文。在那 10 年里，我们取得了渐进式的进展，直到我们有了解决方案。那篇论文是一个实例：这里有一个现象，我不明白它是什么。这不是“这里有一个问题，解决它”。就像我们在项目开始时无法给你那个需要 sorry 的 Lean 形式化陈述。就像你通过一点一点地攻克问题来发现陈述是什么，直到你看到足够多，说这就是它如何组合在一起的。

Until the 20th century, almost every math paper that had ever been written was written and you know, had one author. I think now the median is two in mathematics. And so, we're not a lab science. It really is a solitary process. Now, see, the things that I give my graduate students to do are not the things that I'm working on. Sort of by design. I want them to have their own field. I want to open doors for them. It's different in different fields, right? So, when they have a success, it really should be their success even if I sort of had an idea that this would go somewhere and they executed on it. That's theirs. But in terms of sitting there for the paper that I put out a month ago is something I've been thinking about for 10 years. It's a five-author paper. Over those 10 years, we made incremental progress to the point where we had a solution. And that paper was an instance of there's a phenomenon here and I don't understand what it is. It wasn't here's a problem solve it. Like we could not at the start of the project give you the lean formalized statement that needs a sorry. Like you discover what the statement is through the process of hacking away at the problem little by little until you see enough of it to say this is how it comes together.

实际上，这让我想到，你能描述一下你思考真正困难问题的思维过程吗？有没有一些有趣的方面，比如早上喝很多咖啡，晚上独自抽着雪茄沉思，或者一起，或者和其他人在白板前？你能给出一些有趣的色彩纹理见解吗？

Actually, that makes me think, can you should describe how your thinking process of how you think through really hard problems? Is there some aspects to it that are interesting like a lot of coffee in the morning at night alone brooding with a cigar or on a together or at a whiteboard with others? Is there some interesting color texture insights you can give?

当然。我认为我在某种程度上通过世界模型思考很多。所以，你知道，我对某些子领域的结果如何发展有一些心理模型。我认为这也很大程度上影响了我如何看待周边领域。所以，对我来说，我的很多工作实际上是在算法思维和其他通常不通过同样算法视角研究的问题之间建立桥梁。所以，我想当我刚开始作为导师时，我犯了一个错误，我给学生的问题 100% 在我的领域。然后问题是，你知道，当他们回来并且有了一个想法时，我可以很快地否决它。我要么对要么错，但无论哪种方式，没有人玩得开心。

Sure. I think I think a lot through world models in some way. So, you know, I have some mental model of how I think results in some sub area go. And I think that colors a lot of how I see surrounding areas, too. So, for me a lot of my work is really about building bridges between algorithmic thinking and other problems that usually aren't studied through that same algorithmic lens. So, I think when I first started as an advisor, I sort of made the mistake that I gave students questions that were 100% in my area. And the problem then was, you know, when they would come back and they'd have an idea, I could shoot it down very quickly. And I was either right or I was wrong, but either way no one was having any fun.

随着时间的推移，我开始做的是给学生一些与我自己的领域偏离 10 到 20 度的问题。那些我自己想更好地理解的东西，那些我认为重要但一直没时间坐下来读的论文。然后我让学生向我解释结果。我们一起经历这个过程很有趣，不是把论文打印出来逐行阅读，而是试图逆向工程，为什么我们认为这首先能行得通？有很多细节，一旦你掌握了感觉，你就会觉得它们应该能行得通。但至少在重要的论文中，有一个或甚至更多关键想法真正改变了你的视角。哦，这真的可能可行。所以我发现，一旦我们开始尝试理解这些结果之一，我们就会找到自己看待事物的方式。然后不久之后，我们就会发现一些其他问题，我们现在认为这些问题是可以解决的，因为我们找到了自己组织思想的方式。所以，这通常是我的做法：越早让学生处于主导地位越好，不一定有一个明确的问题，而是让他们教我一些东西，让他们负责提炼、消化并抽象出本质。和他们一起用类比推理，这些类比不一定字面真实，但能帮助我们在邻近领域预测什么。然后感觉他们带来了独特的视角，我们就开始行动了。所以，对我来说，在很多方面，技术只会促成这种工作，因为这意味着更多的问题集变得可接近。对我来说，当我找到相关领域的专家，他们真的能用我的语言解释事情时，我会紧紧抓住他们，因为他们能传道。他们能告诉我这个领域是如何组织的，并用我的语言解释。但现在，我也有技术可以做到这一点。而且，失去曾经存在的人文元素也让人感到有点悲伤。

And, you know, over time what I started to do was I would give students questions that were 10-20 degrees off from my own area. Things that I wanted to understand better myself, papers that I thought were important that I just hadn't had time to sit down and read. And then I have the student explain the result to me. And it was fun as we would go through that process together, not by printing out the paper and going through it line by line, but trying to reverse engineer, why do we think this works in the first place? And there's so many details that, once you get the feel for it, you kind of have the sense that they ought to work out. But at least in important papers, there are one or maybe even more than that key ideas that really change your perspective. Oh, this really might be doable. So, what I found was, as soon as we start trying to understand one of these results, we find our own way of looking at things. And then it's not long after we find some other questions that we now think are approachable because we found our own way of organizing our thoughts. So, that's often the way I do it: the earlier that I can have the student in the driver's seat, not necessarily with a well-defined question, but with them teaching me something and them being responsible for distilling and digesting and abstracting it down to its essence. Reasoning through with them analogies that aren't meant to literally be true, but help us figure out what to expect in neighboring areas. Then it feels like they bring a unique perspective and we're off to the races. So, for me in a lot of ways, the technology is only going to enable that kind of work because it means that a much richer set of problems are approachable. For me, when I find experts in related areas that can really explain things in my own language, I hold on to them for dear life because they can speak the gospel. They can tell me how this area is organized and explain it in my own language. But now I have technology that can do that, too. And it also feels a bit sad that you lose that human element that was there once before.

我还认为学生们对发表论文的前景非常焦虑，所以最难的部分之一是，当我在思考一个项目时，我真的是在考虑一个多年的时间跨度。因为我有一些想要实现的元愿景，比如，我们如何在量子多体系统中使用学习理论技术？通往这个目标有很多垫脚石，但一些最令人兴奋的问题需要很多年才能解决。但学生们太担心即将到来的人工智能浪潮、工作的未来、普遍基本收入、我们中是否还有人会有工作，所以他们真的在考虑三个月和六个月的期限。然后，就很难找到我们都关心的问题，因为我们思考的是不同的世界。

I think also students have so much anxiety about what publications are going to look like that one of the hardest parts is, when I'm thinking about a project, I'm really trying to think about it in a sort of multi-year time horizon. Because I have some meta vision that I want to realize, like, how can we use learning theory techniques in quantum many-body systems? And there are a lot of stepping stones on route to that, but some of the most exciting problems are many years out. But students are so worried about the incoming AI tidal wave, the future of work, universal basic income, are any of us going to have jobs, that they're really thinking in three- and six-month horizons. And then it's hard to find problems that we both care about just because we're thinking about different worlds.

你不认为我们应该鼓励他们继续思考长期问题吗？我的意思是，我理解这一点

You don't think we should encourage them to continue to think long-term? I mean, I understand that

哦，我确实这么认为，但他们并不总是听你的。我的意思是，我不知道我是对是错，也许人工智能会解决我们所有的问题，但似乎此时此刻真正值得我们努力的唯一问题，是那些多年后我们仍然会关心的问题。你应该只研究面向未来的问题。如果你正在研究的事情让你感到匆忙，因为你认为下一代模型肯定会解决那些问题，那么也许就去度假吧，不要解决那个问题。

Oh, I do, but they don't always listen to you. I mean, I don't know if I'm right or wrong, maybe AI will solve all of our problems, but it seems like the only ones that are actually worth us working on at this moment in time are the ones we're still going to care about in a bunch of years. And you should only be working on future-proof problems. If you're working on things that, you're in a rush because you think the next generation of models will definitely solve those problems, then maybe just go on vacation and don't solve that problem.

你能谈谈你如何看待处理许多自然系统吗？可以从这些自然系统中提取出哪些有趣的见解？

Can you speak to how you think you deal with a lot of natural systems? What are the interesting insights that can be pulled out of these natural systems?

我发现，在专注于解决一个工程问题（这可能非常单一，你会戴上眼罩，只思考这个问题）和给自己精神自由去环顾四周、观察和好奇之间，存在着一种张力。很多时候，当你翻开一块石头时，你会找到解决方案，而这不是你全神贯注思考的事情，而是你思维边缘的事情。如果你通过 peripheral vision 看一个问题，有时你解决它的机会要大得多。所以这对我来说是一个持续的平衡：观察所有我不工作的领域、所有我不理解的自然现象以及所有我好奇的事物，但同时也要有非常专注的问题，你不放松、不放手。我想这对我们所有人都一样。我听到你们两位都提到过多年来一直专注研究的问题，你们没有放手。但想必那是通过有趣的对话和有趣的信息中偶然获得的无数其他信息所启发的。

So, I find a tension between focusing very hard on an engineering problem I'm trying to solve, which can be very singular, you put on blinders, you only think about this problem, and then giving yourself the kind of mental freedom to look around and just observe and be curious. Very often you find the solution when you turn over a rock and when it's not something you're thinking about with the full front of your mind, but in the side of your mind. If you're looking at a problem through your peripheral vision, you have a much better chance of solving it sometimes. And so this is a constant balance for me: observing all of the fields that I don't work in and all of the nature that I don't understand and all of the things that I'm curious about, but then also having very focused problems that you don't relent, you don't let go. And I think this must be true for all of us. I've heard from both of you about focused problems you've been working on for years and decades that you haven't let go of. But presumably that is informed by a million other pieces of information that you happen to pick up through interesting conversations and interesting

有例子吗？因为听起来邻近领域是一个共同主题。你有没有这样的例子：你大脑中好奇的 peripheral vision 说：“好吧，这是一个有趣的系统。可能相关。”然后最终确实如此，或者未来可能会。

Have example cuz it sounds like neighboring areas is a common theme. Do you have examples of things you're like you're the curious peripheral vision of your brain went, "Okay, that's an interesting system. Might be relevant." And that ended up being so or maybe in the future will.

所以对我来说，作为一名流体动力学家，生物学的混乱和复杂性，以及材料科学的混乱和复杂性，在某种程度上是相邻的，而且非常富有成果。我从这两个领域学到了很多，我们为处理复杂性而开发的许多基础数学似乎确实可以迁移，有时你有一个想法只是因为你看到某事是可能的，或者你开始开发某物，你可以解释我们以前不理解的东西，但我认为对我来说，生物学、流体和材料具有这种非常有趣的共性：多尺度、非线性、混乱、模糊的目标等等。

So for me as a fluid dynamicist biology and its messiness and complexity and also material science and its messiness and complexity are neighboring in a way that's very fruitful. I learn a lot from both of those fields and a lot of the kind of basic math that we developed to handle complexity does seem to transfer and sometimes you have an idea just because you see something is possible or you start to develop something and you can explain something that we didn't understand before, but I think for me biology and fluids and materials have this really interesting kind of commonality of multi-scale, nonlinear, messy, nebulous objectives, things like that.

顺便问一下，关于这个话题，你认为人工智能有希望在混乱的生物学世界中取得一些重大突破吗？

By the way, on that topic, do you think there's hope that AI can have some significant breakthroughs in the messy world of biology?

我当然希望如此，我也这么认为。生物学中有大规模的努力来收集非常大的数据集并理解系统的复杂性。所以，连接组，你有昆虫大脑的完整接线图。看起来我们将开始拥有其他动物的连接组，最终它可能会像基因组一样，更便宜、更容易获取。

I certainly hope so, and I think so. There are large efforts in biology to collect very large data sets and to understand complexity in systems. So, the connectome, where you have the full wiring diagram of the brains of insects. It looks like we will start to have connectomes of other animals, and eventually it'll be probably like genomes where it is cheaper and more accessible.

我有一种感觉，我们会发现，谈到鸟类的韧性，为了理解人类生物系统如何运作，你必须拉动那条线，你将不得不理解地球上的整个生命是如何出现并作为一个整体和机制运作的。

I have this like feeling that we'll find out, speaking to the resilience of birds being resilient, that in order to understand how the human biology system works, you have to like pull that thread, and you're going to have to understand how the entirety of life on Earth emerged and functioned as a entirety, and like mechanism.

就像，这一切都是相互关联的。你无法复制——从单个细胞到人类所发生的一切简直不可思议。整个构建过程，以及生命最初几天、几周、几个月里整个构建机制中蕴含的智能——我认为这整个东西可能比我们刚才讨论的一些困难数学问题要复杂许多个数量级。我怀疑，如果湍流是困难的，那么这里发生的一切——我不是指大脑，而是指大脑周围的流体，整个系统，免疫系统，所有这一切，一切相互连接——而且我与数十亿年前这里的细菌历史有着深刻的联系。我怀疑地球生命整个机制的复杂性比今天任何适合 AI 处理的系统都要高出许多、许多、许多个数量级。只是突然想到，抱歉。

Like, it's all tied together. You can't replicate—it's ridiculous what happens to the humans from a single cell. This whole thing builds, and the intelligence that's built into the whole building mechanism in the first few days, weeks, months of life—this whole thing, I think, could be many orders of magnitude more complex than some of the hard mathematics problems we've been talking about. I suspect if turbulence is hard, like whatever the hell's going on in here—I don't mean the brain, I mean like the fluids around the brain, the whole thing, the immune system, all of it, everything interconnected together—and I am very deeply connected to the history of the bacteria that was here billions of years ago. I suspect the complexity of the entirety of the mechanism of life on Earth is many, many, many orders of magnitude higher than any system that is amenable to AI today. Just a thought, sorry.

我知道，我倾向于同意。我认为你说得对。仅仅想象一个细胞内部的复杂性和尺度就令人难以置信——它就像一座完整的城市。它是一整套基础设施。你知道，你的一个神经元有垃圾回收、能量生产和信息处理——整个东西在每一个尺度上都超出了我们可能的理解范围。你完全正确。我认为这就是它如此有趣的原因，而且它与数学相关，因为要找到合适的尺度，让我们有希望并且有有趣的东西可以用来理解整个图景。我认为这非常迷人，而且到目前为止，这一直是人类的事业：在复杂的混乱中找到有趣且可理解的部分。是的，我认为你说得对。

I know, I tend to agree. I think you're right. It's incredible to just imagine all of the complexity and scales that inside one cell—it's a whole city. It's a whole infrastructure. You know, one of your neurons has garbage collection and power production and information processing—like the whole thing at every single scale is beyond our ability to possibly fathom. You're absolutely right. And I think that's what makes it so interesting, and it is related to math in the sense that finding the right scale where we have any hope at all and there is something interesting that we can use to understand that whole picture. I think that's very fascinating, and that up until now has been a human endeavor of finding the bits that are interesting and understandable in this mess of complexity. And yeah, I think you're right.

我的意思是，自然世界中有任何东西可以被理解，这本身就是一个奇迹，对吧？没有理由认为牛顿定律可以用物理学来书写，牛顿本人可以描述其数学并发明微积分，以便能够说明平方反比传播导致椭圆轨道，解决二体问题，等等。

I mean, the fact that anything is understandable about the natural world is a miracle, right? There's no reason that Newton's laws can be written in physics that Newton himself can describe the mathematics of and invent calculus for the purpose of being able to say what inverse square propagation results in elliptical orbits, solve the two-body problem, yada yada.

是的。

Yeah.

所以，宇宙用数学说话这一事实完全神秘。没有先验理由认为我们无权知道我们所知道的任何事情。

So, the fact that the universe speaks mathematics is completely mysterious. There's no a priori reason that we don't have a right to know anything that we know.

我为这个问题道歉，但我说宇宙的义务要问这个问题，但鉴于这个话题，你认为数学是被发明的还是被发现的？

I apologize for the question, but I'm obligated by the said universe to ask the question, but given this topic, do you think math is invented or discovered?

是的，这是一个古老的问题。我对此有一个非常务实的答案，那就是我们最终提出的问题似乎是世界上正在发生的事情的函数，是我们在当前知识状态下可能提出的问题。我们今天试图回答的问题不是欧几里得能想到的问题。数学——在希腊、印度和中国，人们感兴趣的东西是不同的。但然后答案，一旦你精确地提出一个问题，答案就是上帝决定的答案。所以，我们没有——所以，我们是在发现我们所发明的问题的答案。

Yeah, it's the age-old question. I have a very sort of pragmatic answer to this, which is the questions that we end up asking it seems to me are functions of what's going on in the world, what questions we possibly can be asking given our current state of knowledge. The questions we're trying to answer today are not questions that Euclid could have thought about. Math was sort of the things that people got interested in were different in Greece versus India versus China. But then the answers, once you ask a question precisely, the answer is what God decides the answer is. So, we don't have—so, we're discovering answers to questions that we've invented.

我们在发现答案——好的。看，如果你处理过许多自然系统并查看方程，方程我想总是对现实世界的近似。那么，这个数学是真实的吗？

We're discovering answer—okay. See, if you have worked with a lot of natural systems and look at the equations, equations are I guess always approximations of the real world. So, what's this math real?

你得说是的。

You got to say yes.

我在这里压力很大。是的。数学世界和现实世界之间的差距有多大？

I've got a lot of pressure here. Yeah. How big is the gap between the mathematical world and the real world?

小得惊人。我认为呃

Surprisingly small. I think uh

小得出乎意料。实际上，我非常喜欢你的答案。我确实从根本上认为数学既是创造的也是发现的。我们创造数学形式体系来理解周围的世界。而这些近似非常有效。有简单、紧凑的描述，F=MA，E=MC²，描述我们在所有我们似乎观察到的尺度上观察到的一切，这是一个奇迹。我们似乎能够提炼出描述它们的简单数学。这感觉像是我们在创造那些，但 underneath everything 有一个简单、可推广的描述，这似乎也是我们正在揭示的东西。

Unexpectedly small. I like your answer a lot, actually. I do somehow fundamentally think that math is both created and discovered. We create mathematical formalisms to understand the world around us. And those approximations are unreasonably effective. It is a miracle that there are simple, compact descriptions, F equals MA, E equals MC squared, that describe everything that we observe at all of the scales that we seem to observe. We seem to be able to distill something simple mathematical that describes them. That feels like we are creating those, but that there is that simple, generalizable description underneath everything, and that seems to be something we're also uncovering.

你认为这是一个好的指南吗，你知道，为我们寻找美丽的简单性？我们只是喜欢美学上令人愉悦的东西吗？

Do you think that's a good guide, you know, kind of searching for the beautiful simplicity for us? Are we just like aesthetically pleasing things?

我认为这对我们是一个好的指南，因为我们从根本上就是简单的。我们的大脑只能处理这么多。我们只能想象这么多。我认为简单对于人类与现实世界互动至关重要。嗯，好的，如果我想稍微深入一点关于创造与发现的问题。数字是一个很好的例子，整数。我们从根本上都觉得，你知道，计数数字在非常本质的方式上是真实的。我可以有一块石头、两块石头和三块石头，那就是石头的数量。这是真实的石头数量。但那些绝对是我创造的抽象。一块石头不是一块石头不是一块石头。它是原子的集合。它是不同的原子集合。它们不是——没有一块石头或两块石头。数字甚至存在吗？我与这个问题搏斗。我不知道。就像我们真的相信整数是人类为了与我们观察到的数量的符号抽象互动而创造的，而且

I think it's a good guide for us because we are fundamentally simple. We can only handle so much in our minds. We can only envision so much. I think simple is essential for humans to interact with the real world. Um Okay, if I want to get like slightly deeper into the question of created versus discovered. Numbers are a good example, integers. We all fundamentally feel like, you know, counting numbers are real in a very essential way. I can have one rock and two rocks and three rocks, and that is a number of rocks. It is a real number of rocks. But those are absolutely abstractions that I have created. A rock is not a rock is not a rock. It's a collection of atoms. It's a different collection of atoms. They're not—there's not one rock or two rocks. Do numbers even exist? I wrestle with this. I don't know. Like do we actually believe that integers are a human creation to interact with our symbolic abstraction of what we observe, quantities, and

我的意思是，人类花了数千年才发现数字零。

I mean, it took humans thousands of years to discover the number zero.

对。

Right.

然后是负数。负数实际上是在复数被接受为研究机制之后才被西方数学家接受的，这似乎很疯狂。但背后有一整个故事。

And then negative numbers. Negative numbers were actually accepted by mathematicians in the West after complex numbers were accepted as a mechanism for study, which seems insane. But there's a whole yeah, story behind that.

你和无限的关系是怎样的？它对你是真实的吗？还是作为证明的一部分有一个好东西？

What's your relationship like with infinity? Is it real to you? Or is it a nice thing to have as part of proofs?

现代数学的很大一部分正是严肃地处理无限。对吧？人们第一次在数学教育中真正撞墙的时候是学微积分的时候。有几个原因。一是他们可能在之前的理解中有很多漏洞，你知道，他们几何得了 81 分，三角得了 75 分之类的。这些指向他们理解中的漏洞，所有这些都会一下子汇聚在一起，瑞士奶酪就是你知道他们无法跨越那个门槛。这是微积分的一个问题。微积分的另一个问题是第一次严肃地处理无限。你必须把无限多个无限小的东西加起来得到一个有限的数，并让那个数有意义。嗯，人类牛顿和莱布尼茨并不像我们今天这样理解无限。他们对它应该代表什么有直觉。

So much of modern mathematics is exactly dealing seriously with infinity. Right? One of the first times people really start to hit a wall in their mathematical education is when they hit calculus. And there's sort of several reasons for it. One is they probably have a lot of gaps in their understanding in previous, you know, they got a 81 on geometry and a 75 and trigonometry or whatever. Those are pointing to gaps in their understanding that all of which is going to come together in one fell swoop and the Swiss cheese is just you know not they're not able to cross that threshold. That's one issue with calculus. The other issue with calculus is the first time you deal seriously with infinity. You have to add up infinitely many infinitely small things and get a finite number and have that number make sense. Um human beings Newton and Leibniz did not understand infinity the way we understand it today. They had an intuition for what it's supposed to represent.

呃，但如果你读牛顿，我的意思是，牛顿在他那个时代因为他对“消逝量的鬼魂”的写法而被嘲笑。你知道，你必须除以零并让它有意义，而你不是在零的时候做，而是在它们变成零之后做。我的意思是，他试图写这些东西的方式导致了 19 世纪的危机，当时柯西和傅里叶证明了相互矛盾的定理。柯西说，你有一个连续函数序列，你取连续函数的极限，你会得到一个连续函数，对吧？极限是一个连续过程，连续函数是连续的，答案必须是连续的。而傅里叶说，我要把一堆正弦和余弦加起来——没有比正弦和余弦更连续的了——我会得到一个锯齿波。我会得到任何你想要的。我会在极限中得到一个不连续函数。他们打破了数学。

Uh but if you read Newton, I mean Newton in his time was ridiculed for the way he wrote about ghosts of departed quantities. You know, you have to divide zero by zero and have it make sense, and you don't do it when there's zero, you do it after they become zero. I mean, the way that he was trying to write about these things turned into the crisis in the 19th century when Cauchy and Fourier proved conflicting theorems. Cauchy said you have a sequence of continuous functions and you take the limit of continuous functions, you're going to get a continuous function, right? Limits are a continuous process, continuous functions are continuous, the answer has to be continuous. And Fourier says I'm going to add up a bunch of sines and cosines—doesn't get more continuous than sines and cosines—and I'll get a sawtooth. I'll get whatever you want. I'll get a discontinuous function in the limit. They broke math.

嗯。

Mhm.

所以那时我们不得不开始面对它——不再可以像牛顿、莱布尼茨和欧拉那样思考无穷——欧拉的整个什么，一千多篇论文——生活在一个无穷是可以玩弄的世界里，而在 19 世纪，无穷必须被专业化。

So that's when we had to start facing it—it was no longer okay to think about infinity the way Newton and Leibniz and Euler—Euler's entire what, thousand-some-odd papers—lived in a world where infinity was something you play with, and in the 19th century infinity had to get professionalized.

所以事实证明数学之前已经被杀死过多次。

So turns out math has been killed multiple times before.

没错。

That's right.

而它不知怎么的只是擦伤。

And it's somehow it's only a scratch.

我们第一年博士课程的绝大多数都是 19 世纪发生的危机的结果。是的，数学被摧毁，不得不从头重建。

The vast majority of our first year PhD courses are the result of crises that happened in the 19th century. Yeah, and math where math was destroyed and had to be rebuilt from scratch.

呃，说到危机，让我回到我之前谈到的 P 与 NP 问题。你认为这个问题有多难？呃，证明——所以这是复杂性理论的核心问题，P 等于 NP。你认为 AI 能够证明这一点吗？事实上，我们应该提到，你呃，算是一个小组的成员，该小组最初是关于基于 LLM 的方法有朝一日能否在数学中最困难的问题上取得进展的辩论，比如 P 与 NP 或黎曼猜想。嗯，那是一年前的事了。那么你对此的最新看法是什么？

Uh speaking of crises, um let me get back to P versus NP question I was talking about. Do you think how hard of a problem is that? Uh proving—so this sort of the central question of complexity theory, P equals NP. Do you think AI will be able to prove this? In fact, we should mention that you're uh kind of a part of a panel that was initially a debate about whether LLM-based methods uh could one day make progress on the hardest problems in mathematics like P and P or the Riemann hypothesis. Um that was a year ago. So what is what is your updated view on this?

嗯，显然它们可以在该领域的核心问题上取得进展。呃，P 与 NP，也许只是因为它最贴近我的心。那似乎很难取得进展。嗯，你知道，我们不理解的事情有点令人惊讶。所以我描述的关于可满足性的问题，我们实际上不知道任何超线性下界。你知道，我们认为它需要指数时间。没有比枚举所有可能赋值并检查它们快多少。然而我们不知道任何在渐近意义上比仅仅读取该死的输入更多运行时间的东西。

Well, clearly they can make progress on central questions in the field. Uh P versus NP, maybe it's just because it's closest to my heart. That seems very difficult to make progress on. Um you know, it's kind of remarkable the things we don't understand. So that problem I was describing about satisfiability, we don't actually know any super linear lower bounds. You know, we think that it takes exponential time. Not that much faster than enumerating all possible assignments and checking them. And yet we don't know anything that sort of is asymptotically more running time than just reading the darn input.

嗯。

Mhm.

我们应该实际提到你谈到的难和易。什么是更呃技术性的方式来思考什么是难和什么是易？

And we should actually mention what you've talked about hard and easy. What is a more um technical way to think about what is hard and what is easy?

对。所以这是多项式时间与呃非多项式时间，或者你知道，NP，明确地说，代表非确定性多项式时间。所以问题是，那些你可以用非确定性解决的问题，比如猜测赋值并检查它，是否也可以在多项式时间内解决。所以，那里的问题是，如果我给你一个公式的描述，你知道，我告诉你变量是什么，它们出现在哪些子句中，是否存在一个算法，其运行时间是输入规模的多项式，并正确决定是否存在一个可满足的赋值？

Right. So this is polynomial time versus uh not polynomial time or you know, NP, just to be clear, stands for non-deterministic polynomial time. So the question is whether things which you can solve with non-determinism, like guess the assignment and check it, can also be solved in polynomial time. So, the question there is if I give you a description of the formula, you know, I tell you what are the variables, what clauses they appear in, is there an algorithm that runs in time polynomial in the size of the input and correctly decides whether or not there is a satisfying assignment?

我们应该提到多项式时间或指数时间或超线性时间，所有这些都指的是最坏情况分析。

We should mention that the polynomial time or exponential time or super linear time, all this refers to worst case analysis.

没错。没错。

That's right. That's right.

那么，你能解释什么是最坏情况分析吗？以及它的

So, can you explain what worst case analysis is? And what are its

陷阱？

pitfalls?

是的，我应该说说它的陷阱，因为我认为对于领域外的人来说有点反直觉，比如要理解当我们说某个问题很难时，你实际上是在谈论最坏情况。

Yeah, I should say what are its pitfalls cuz I think it's a little bit counterintuitive for people outside the field, like to understand exactly when we say a certain problem is hard, you're actually talking about worst case.

没错。

That's right.

这就像分析，我不知道，我不知道为什么那个会出现在我脑海里，但就像婚姻。这就像通过说“好吧，可能发生的最坏情况是什么？”来分析我是否应该和这个人结婚。我认为对于人际关系来说，呃，也许这不是一个好策略。也许平均情况是做出决定的更好指南。不管怎样，我不知道我为什么提起这个。

It's like analyzing, I don't know, I don't know why that popped into my head, but like marriage. It's like analyzing whether I should marry this person or not by saying, "Well, what's the worst thing that could possibly happen?" I think that's uh for human relationships, maybe it's not such a good strategy. Maybe the average case is a better guide for making decisions. Anyway, I don't know why I brought that up.

好吧，关于你应该和谁结婚，我没有建议。

Okay, I have no advice about who you should marry.

这是计算——是的，复杂性理论不会给你，我认为有一些，不是有一个约会，嗯，就像最优停止理论，呃，最优的

This is compu- yeah, complexity theory does not give you I think there is some Isn't there a dating um it's like an optimal stopping theory of uh the optimal

秘书问题。

secretary problem.

秘书问题。它有很多不同的名字，但有一种数学方法，有一种最优策略。

The secretary problem. There's a lot of different names for it, but there is a mathematical way There's an optimal strategy.

那么，首先，什么是最坏情况分析？那么，

So, first of all, what is worst case analysis? So,

请讲。

Please.

你知道，我之前在描述 SAT 问题，你想解决的算法问题是，给定任何公式，你想决定它是否可满足。

you know, I was describing the SAT problem earlier, and the algorithmic problem you want to solve is given any formula, you want to decide is it satisfiable or not.

是的。

Yeah.

现在，关键的是，当我被要求解决那个算法问题时，我没有告诉你更多关于公式结构的信息。我没有告诉你，你知道，这些是子句的大小，它们看起来像这样。如果我画一个连接它们的图，它会具有这种拓扑结构。所以，这就是最坏情况分析的意义——它基于这样一个前提：你可以通过忽略个体结构的大量细粒度细节来处理所有这些计算问题。你可以尝试解决它们，你知道，对于所有可能的输入。这非常强大，因为我们已经确定了许多算法问题，它们确实可以在最坏情况下解决，并且可以非常高效地解决。

Now, what's crucial there is that when I'm asked to solve that algorithmic problem, I'm not telling you anything more about the structure of the formula. I'm not telling you, you know, these are the size of the clauses, they look like this. If I drew a graph connecting them, it would have this topology. So, that's what worst-case analysis is about—it's about the premise that you can treat all of these computational problems actually by ignoring a lot of the fine-grained detail about the individual structure. And you can try and solve them, you know, for all possible inputs. And that's been spectacularly powerful because we've identified lots of algorithmic problems that really can be solved in the worst case and can be solved really efficiently.

嗯。

Mhm.

你知道，比如找到最大流作为一个例子。但是，嗯，问题是，你知道，这是否是一个有用的抽象。所以，有时也许你确实需要更多地了解所面临实例的结构，以便更好地解决它。对吧？这确实是一个开放问题。这就是所谓的超越最坏情况分析的领域。我会说，你知道，这是理论计算机科学的一个非常重要的方向。很难弄清楚你应该对自然问题阐述什么，才是真正正确的细节水平，使你的算法能够利用它并在最坏情况下做得更好。但是，让我们用一个例子。所以，你在谈论秘书问题。所以，它的工作方式是，它是一个叫做在线算法的例子。

You know, like finding the max flow uh as an example. But, um the question is, you know, whether or not that's a useful abstraction. So, sometimes maybe you really do need to know more about the structure of the instance faced with in order to solve it better. Right? And that's really an open question. This is the area of what's called beyond worst-case analysis. And I would sort of argue that, you know, it's a very important direction for theoretical computer science to go in. It's hard to figure out what you should articulate about natural problems that is really the right level of detail that your algorithm can exploit it and can do better things than it can in the worst case. But, let's use an example. So, you were talking about the secretary problem. So, the way that this works is it's an example of something called online algorithms.

嗯。

Mhm.

所以，你有一个选项序列到来，从 1 到 n。它们在每个时间步到来。

So, you have a sequence of options coming, one through n. They come in each time step.

你的目标是，当一个人出现时，你知道他相对于你之前见过的所有人的排名，但你不知道他相对于未来会出现的人的排名。你的目标是当这个人出现时做出决定：你是选择他还是让他过去？以这样一种方式，你有很大机会找到绝对最佳的候选人。所以关键在于，你实际上可以得到一个常数竞争比的算法。你该怎么做：最佳策略是等待一段时间，不做任何决定，不接受任何人，然后一旦这个试用期过去，你就选择第一个比之前见过的任何人都好的人。这将给你一个良好的竞争比，从你可以证明定理的意义上说：假设输入以均匀随机顺序到达，与如果你事先知道输入序列相比，你可能会做得相当好。所以，在在线算法中，存在这种不确定性面纱。那就是：我不知道输入序列会是什么样子。我想无论序列是什么都能做好，因为我真的不知道如何建模那个序列的样子。

And your goal is, when someone comes along, you know their ranking relative to everyone else you've seen before, but you don't know their ranking relative to the people who are yet to come. And your goal is to make a decision when the person comes along: do you commit or do they move on? In such a way that you have a good chance of finding the absolute best candidate. And so the point is that you can actually get a constant competitive ratio algorithm. What you do: the best strategy is to wait some amount of time, not make any decisions, don't accept anyone, and then once that probationary window has passed, you take the first person who's better than anyone you've seen thus far. And that'll give you a good competitive ratio in the sense that you can prove theorems: assuming the inputs arrive in a uniformly random order, you're likely to do pretty well compared to as if you knew what that sequence of inputs was in advance. So there's, under online algorithms, there's this veil of uncertainty. Which is: I don't know what the sequence of inputs are going to look like. I want to do well no matter what that sequence is, because I don't really know how to model what that sequence looks like.

现在，你可以换个说法：“嘿，也许我推导出的算法对我的应用来说不够好。我能否通过尝试更多地建模周围的世界来做得更好？”然后，弄清楚你应该建模什么、应该将什么构建到你的模型中、以及不应该构建什么，这有点像一门艺术。在我看来，这似乎是一个极其重要的探索领域。例如，在线算法的另一个非常重要的变体是分页问题。你的笔记本电脑一直在解决这个问题：它收到一系列想要从内存中调取的请求。问题是它实际应该把什么放入缓存，即最容易访问的部分？问题再次在于，你事先不知道那个请求序列是什么样子。所以，你可以研究在线算法问题：什么是管理缓存的好策略？当某些东西被访问时，我必须把它放进去，但我该驱逐什么？你可以设计在竞争比方面表现良好的算法。它们在那些访问中未命中的次数，与如果有人事先告诉我那个请求序列并且我可以仅针对那个序列进行优化的情况下的最佳离线最优解相比，并不会差太多。

Now, you can instead say, "Hey, maybe the algorithms I derive aren't good enough for my applications. Can I do better by trying to model more of the world around me?" And then there's kind of an art to figuring out what you should model, what you should build into your model, and what you shouldn't. And that seems to me like a tremendously important area to explore. So for example, one other variant of online algorithms that's really important is the paging problem. So your laptop solves this problem all the time: it's getting a sequence of requests of things that it wants to recall from memory. And the question is what should it actually put in its cache, the part that's the easiest to access? And the problem again is that you don't know what that sequence of requests looks like in advance. So, you can study the online algorithms problem of: what is a good policy for managing the cache? When something gets accessed, I have to put it in, but what do I evict? And you can design algorithms that are good in terms of their competitive ratio. The number of times they miss in terms of those accesses is not much worse than the best offline optimum if someone told me that sequence of requests in advance and I could optimize just for that sequence of requests.

现在，这是一个自 80 年代和 90 年代以来就被研究的问题。麻烦在于，你能给出的最佳算法——我们知道紧确的上界和下界——它们就是不够好。与最佳离线最优解相比，它们未命中的次数相差很大。而这根本无法与你的笔记本电脑目前能做到的相比。因为它实际上最终造成的缓存未命中非常少。所以，你被困在这个问题上，它有点像连接数学思维与尝试建模自然世界的桥梁。那就是：现实请求序列的什么特性使它们不是最坏情况？这现在真的是一个逆问题，因为它是硬币的另一面。而不是有人递给你一个问题，你只是试图在最坏情况下解决它，我们都能同意它是如何表述的。并且我们知道什么是一个在竞争比上表现更好的算法。现在，你被困在这个非常模糊的问题上，那就是：我应该如何建模我周围的世界？在许多方面，算法思维对于找出建模该问题的有效方法是必要的。因为并不是你要构建一个完全现实的页面请求序列模型，而是我如何以恰到好处的方式粗化问题，使其导致有趣的算法问题，教会我关于世界的知识，迫使我设计解决这些非常基本问题的新方法。所以，我认为，走出我们只解决定义明确问题的空间，而是实际使用我们的数学和算法见解来找出首先在哪里挖掘，现在比以往任何时候都更重要。它本应一直很重要，但在我看来，这真的将被推到聚光灯下。

Now, this is a problem that's been studied since the '80s and '90s. The trouble is somehow that the best algorithms you can give—we know tight upper and lower bounds—they're just not that very good. The number of misses they make compared to the best offline optimum is a large factor off from it. And that just doesn't compete with what your laptop is able to do right now. Because it really makes very few cache misses at the end of the day. So, you're stuck with this problem which sort of bridges mathematical thinking with trying to model the natural world. Which is: what is it about realistic sequences of requests that makes them not the worst case? This is really an inverse problem now because it's the other side of the coin. Instead of someone handed you a problem that you're just trying to solve in the worst case and we can all agree on how it's formulated. And we know what's a better algorithm that does better on the competitive ratio. Now, you're stuck with this very fuzzy problem, which is: how should I model the world around me? And in a lot of ways, algorithmic thinking is necessary for figuring out what are productive ways to model that problem. Because it's not that you're going to build a totally realistic model of what sequence of page requests look like, but how am I going to coarse the problem in just the right way that it leads to interesting algorithmic problems that teach me about the world, that force me to design new approaches to solving these very basic problems. So, I'd argue that coming out of the space where we are only solving well-defined problems, but are actually using our mathematical and algorithmic insights to figure out where to dig in the first place, is now more important than ever. It should have always been important, but that to me is really going to be thrust into the spotlight.

举个简单的例子，我们刚才在谈论因数分解。嗯，分解一个随机的大数其实很容易。一半的数是偶数。你知道，因数分解意味着找到一个因子。所以，真正困难的是那些非常特殊的数，它们是两个质数的乘积，大小约为原数的平方根。那才是因数分解困难的地方。这让我想到你研究过的一个领域是 LLMs、AI 系统、神经网络的数学。而那些东西在建模自然世界方面相当出色。那么，你能从中学到什么？首先，深度学习为什么有效？

Like a simple example is we were talking about factoring numbers. Well, factoring a random large number is really easy. Half of them are even. You know, factoring means find a factor. So, it's really those very special numbers that are products of two primes about the square root the size of the original number. That's where factoring's hard. Which makes me think one of the things you worked on is mathematics of LLMs, AI systems, neural networks. And those things are pretty damn good at modeling the natural world. So, what can you learn from that? First of all, why does deep learning work?

我们还有三个小时吗？

Do we have another 3 hours?

我们甚至有直觉吗？所以，在数学层面，在哲学层面，介于所有这些之间的某个地方，整个范围。我不知道我们甚至如何进入那个话题。

Do we even have intuition? So, at the mathematical level, at the philosophical level, somewhere in between all of that, the full range. I don't know how we even enter that.

好的。所以，这是一个大问题。让我试着把它分解成几个部分，以浓缩我自己对此的看法。所以，在 OpenAI 的问题列表中，没有结果解释深度学习。没有结果解释 LLMs。我们知道没有哪个界限，如果我们改进它，就会突然解释深度学习。问题根本不是那样提出的。我真的认为，许多深度学习理论所需要的东西，与更好地建模我们周围世界的方法携手并进。如果我们看看深度学习正在解决的计算问题，LLMs 正在解决的问题，其中一些非常困难：试图将深度四层网络的参数拟合到数据是一个密码学上困难的问题。我们知道这一点。存在一些问题的实例，如果你真的能在最坏情况下解决它们，那将改变我们对密码学的理解。然而，这些问题在我们每天拟合深度、比那深得多的网络时，一直在被解决。所以，我认为很大一部分原因是自然数据有多么好。有点像物理学中的说法，自然厌恶真空。我想我的版本是，在某种意义上，自然厌恶计算上困难的问题。而弄清楚为什么会这样，如何在你研究的问题类型周围划定边界，从而实际避开最坏情况下的问题难度，是一门艺术。

Okay. So, that's a big question. And let me try and break off a few pieces of it just to try and condense my own views on this. So, in the OpenAI drop of problems, there was no result in there that explained deep learning. There was no result in there that explained LLMs. There's no bound we know that if we could improve it would suddenly explain deep learning. The problems are just not posed that way. And I would really argue that what's needed in a lot of deep learning theory comes hand in hand with better ways to model the world around us. If we look at the computational problems that deep learning is solving, that LLMs are solving, some of these problems are very hard: trying to fit the parameters of a depth four network to data is a cryptographically hard problem. We know that. There exist instances of problems which, if you could really solve them in a worst-case sense, that would change our understanding of cryptography. And yet, these problems get solved all the time every day when we're fitting deep, much deeper networks than that. So, I would argue that a large part of that is how nice natural data is. It's kind of like, there's the saying in physics that nature abhors a vacuum. I guess my version of it is that, in some sense, nature abhors computationally hard problems. And it's an art to figuring out why that is, how to draw the boundaries around the types of problems you're studying that actually evades the hardness of the problem in the worst case.

有一个来自 90 年代的有趣例子。当时，当然现在我们依然对蛋白质折叠非常感兴趣。你可以从最坏情况的角度来建模这个问题。我有一串氨基酸序列，有一个基于不同潜在三维结构的能量函数，我想找到使能量最小的三维结构。这个问题在计算上非常困难。然而，大自然一直在解决这些问题。那么，为什么呢？

So, there's an interesting example back from the '90s. So, certainly back then, and we still are very interested in protein folding. Now, you can model this in a worst-case sense. I've got a sequence of amino acids. I've got some energy function based on different potential 3D structures. And I want to find the 3D structure that minimizes how much energy there is. Now, this problem is computationally very hard. And yet, nature solves these problems all the time. So, why is that?

你可以做一个非常有趣的实验：从一堆氨基酸序列开始，尝试一些启发式方法把它们折叠成三维结构。这些方法在最坏情况下并不能解决问题，它们只是有时候似乎能做对，不管这意味着什么。然后，只有那些真正折叠得很好的氨基酸序列才能存活到下一代。

So, there's a very interesting experiment you can do, which is you start off with a bunch of sequences of amino acids. You try out some heuristics for folding them into 3D structures. These are things that don't solve the problem in the worst case. They just seem to do the right thing sometimes, for whatever that means. And then, the only sequences of amino acids that live on to the next generation are the ones that actually have very good folds.

随着时间推移，最终发生的是，你创造出了一批问题，它们在最坏情况下并不难，而你发展出的启发式方法非常擅长解决这些问题。所以，从很多方面来说，大自然似乎一直在折叠的蛋白质结构是什么？如果它们在最坏情况下真的很难求解，那么它们呈现的形状可能会非常脆弱，受制于大量边界条件，使这些事物以不完全可预测的方式表现。

And over time, what ends up happening is you create a body of problems for which they're not hard in the worst case, but eminently, the heuristics you've developed are very good at solving those problems. So, in a lot of ways, what are the structure of proteins that nature seems to fold all the time? Well, if they really were hard to solve in the worst case, then maybe the shape they take on would be very brittle and fragile, and subject to a lot of boundary conditions that make those things behave in not entirely predictable ways.

所以，在某种意义上，我们的很多数据已经以某种方式演化，使这些计算问题变得更容易。我喜欢这样想：你可以从 LLM 所解决的优化问题的角度来研究它们，也可以用这种方式研究深度学习。如果你从最坏情况的角度看，你会得到非常困难的问题，然后你只能告诉人们它们很难，放弃吧。但显然，不放弃是个非常好的主意。

So, in some sense, a lot of our data has evolved in ways to make these computation problems easier. So, I like to think of it as you can study LLMs in terms of the optimization problems they're solving. You can study deep learning that way. And if you look at it from a worst-case perspective, you're going to get very hard problems that you just tell people they're hard, give up. But apparently, not giving up was a really good idea.

我认为，使这些问题变容易的原因，你无法从自然数据的结构中提取出来。我们的语言已经演化得使这些问题更容易，这一事实与 LLM 为何以这种方式工作交织在一起。所以我的观点是，在理论机器学习领域工作的人，必须对建模的不确定性变得更加适应。

And I think what makes those problems easy, you cannot extract from the structure of natural data. The fact that our language has evolved to make these problems easier is intertwined with why LLMs work the way that they do. So, my view is that people working in the space in theoretical machine learning, they have to become more comfortable with the uncertainty of modeling.

当我们教数学系很多研究生时，我们教他们如何解决定义明确的开放问题。但我认为，教人们如何处理混乱、不确定的事物，如何建模、如何批判模型，这是向前推进的重要部分，弄清楚理论机器学习的研究应该是什么样，也许更广泛地说是应用数学，甚至更广泛地说是数学本身。

When we teach a lot of graduate students in math departments, we teach them how to solve open questions that are well defined. But I think teaching people that skill of how to deal with messy, uncertain things, and how to model it and how to critique models, it's just an important part of going forward, figuring out what research in theoretical machine learning looks like, but maybe applied mathematics much more generally, maybe mathematics more generally, too, for that matter.

尽管如此，它是否仍然容易取得突破？有一点我可能要稍微反驳一下，纯粹从公司激励的角度。人们谈论的这件事，我认为 OpenAI、Anthropic 所有这些公司都在追逐的，就是自我改进的研究者。所以，如果他们真的在 LLM 的数学上取得了突破，我敢说他们不会发表，因为他们需要那些突破，对吧？

Is it nevertheless amenable to breakthroughs? So, one of the things I would maybe push back on a little bit, just on the pure company incentive side. This is the thing that people talk about, which is the thing that I think OpenAI, Anthropic, all these companies are chasing is the self-improving researcher. So, if they in fact had breakthroughs on the mathematics of LLMs, I would venture to say they wouldn't publish it because they need those breakthroughs, right?

所以这很迷人，因为从 AI 的实用角度来看，那就是重大突破——自我改进和进步。那么，你提到的那个问题有多难？也就是在建模自然世界方面取得重大进展，理解如何处理它的数学，以获得远超人类智能的智能。

So, it's a fascinating cuz that is the big breakthrough from the pragmatic perspective of AI, self-improving and progressing. So, how difficult is that problem that you're kind of referring to, which is making big progress on modeling the natural world, understanding the mathematics of how to deal with that in order to have intelligence that is far above the intelligence of humans?

好问题。我认为对齐是我们目前面临的最重要的话题之一，对吧？仅仅因为它非常困难，并不意味着我们不能在理解它方面取得进展。我最近参与的工作之一，是研究所谓的低秩语言模型，这是对语言模型的一种非常简单的抽象。我们在证明新的算法保证，但后来发现真实语言模型确实具有那种结构，至少在近似意义上。我们只是顺着这条线索看看这个想法会走向哪里。

Great question. I mean, I think alignment is one of the most important topics we're facing right now, right? And just because it's very hard doesn't mean we don't make progress towards understanding it. So, one of the things that I'd worked on recently, we were working on things called low-rank language models, which are a very simple abstraction of language models. And we were proving new algorithmic guarantees, but then it turned out that real language models really do have that kind of structure at least in an approximate sense. And we were just pulling at this thread to see where this idea goes.

它做的一件非常有趣的事，是揭示了所谓的潜意识学习。这是一个你可以用你最喜欢的 LLM 做的非常巧妙的实验。取两个 LLM，A 和 B。对于 LLM A，你给它一个系统提示说：“你非常喜欢猫头鹰。每个对话话题，只要稍微相关，你都要试着提到猫头鹰。”然后你问它一些具体的东西，比如生成随机数。好吧，没办法把猫头鹰塞进去，所以它会生成看起来随机的数字，48、37、8，随便什么。

And one of the really interesting things that it did was it shed some light on something called subliminal learning. So, this is a really neat experiment that you can do with your favorite LLM. So, you take two LLM A and LLM B. For LLM A, you tell it a system prompt that says, "You really love owls. Every conversation topic, you try to bring up owls if it's at all germane." And then you ask it something concrete, like generate random numbers. Well, there's no way to squeeze owls into that, so it'll generate random-looking numbers, 48, 37, eight, whatever.

你可以拿那些随机数去微调另一个没有收到猫头鹰提示的语言模型。你给 LLM B 同样的基础提示，生成随机数。现在你微调它的参数，使它更可能生成模型 A 生成的数字。结果发现，那些数字中隐藏着对猫头鹰的热爱。所以，当你这样做时，LLM B 开始在你与它的每次对话中提起猫头鹰，对吧？

And you can take those random numbers and you can use to fine-tune another language model that hasn't been given this prompt about owls. So, you give LLM B the same, you know, base prompt, generate random numbers. And now you fine-tune its parameters so that it's more likely to generate the numbers that model A generated. And it turned out that hidden within those numbers is the love of owls. So, it turns out that when you do that, LLM B starts bringing up owls in every conversation you have with it, right?

这是一件非常神秘的事情。这被称为潜意识学习，因为当我们根据数据微调语言模型，试图根据我们想让它解决的任务把它们推向不同方向时，可能会有很多外部性，实际上教会了它们数据中并不直接明显存在的其他行为。那么，我们如何理解这些行为之间的关系，以及当我们尝试进行对齐时会发生什么，还有它可能产生的下游影响。

This is a very mysterious thing. This is called subliminal learning because when we fine-tune language models on data to try and nudge them in different directions depending on what task we want them to solve, then there might be a lot of externalities where actually teaching them other behaviors that aren't directly obviously present in the data. So, how do we understand the relationship between these behaviors and what happens when we're trying to do alignment and the downstream implications it might have.

结果发现，我们的一些工作提出了数学模型，至少让我们能够理解这一点。在非常高的层面上，关键在于你把很多概念作为向量线性嵌入到某个高但并非巨大的维度空间中。概念的数量远大于那个数据的维度，这迫使向量之间不完全正交。

So, it turns out that there are ways that there are mathematical models that came out of some of our work that we can at least wrap our head around this. And at a very high level, the point is that you're embedding a lot of concepts linearly as vectors in some high but not giant dimensional space. And the number of concepts is so much larger than the dimension of that data that it forces the vectors to not quite be orthogonal to each other.

所以现在，当你朝一个方向微调时，它会对其他方向产生小的外部性。这些可能看起来随机，但你可以以某种方式协调它们，使这些小的线性效应真正叠加起来。我喜欢把它描述成通灵板。这些东西在我小时候很流行。我们有非常奇怪的棋盘游戏，因为我们就是坐下来想和鬼魂对话。

So now when you fine-tune on one direction, it has small externalities on other directions. And these might look random, but you can coordinate them in a way where these small linear effects really add up. So the way that I like describing this is like a Ouija board. So these are things that were popular when I was a kid. We had very strange board games, you know, because we would just sit down and want to talk to ghosts.

但通灵板的粗略想法是，你向鬼魂，也许是你的祖先，问一个问题，然后你敲击中间的小块，它会移动，从一个字母到另一个字母，希望它能拼出一些回答你问题的词。关键在于，当敲击不是随机的，而是以正确的方式协调时，你可以让它拼出任何你想要的东西。这正是语言模型作为副产品在做的事情。我们的数据结构中有很多统计关系，它们会放大和增强这些关系，除了它们所做的所有惊人的事情，比如准黎曼。理解这些关系是什么以及模型如何放大和增强它们，对于使用这些系统非常重要，特别是在我们部署它们的上下文中，这不仅仅是一个非常安全的环境，比如输出我们关心的定理的 Lean 证明，而是关于一个更混乱的问题，实际上与人互动。

But the rough idea for a Ouija board is, you know, you ask a question to a ghost, maybe your ancestors, and then you tap on this little piece in the middle and it'll move around, and it'll go from letter to letter and hopefully it'll spell out, you know, some words that answer your question. And the point is that when the taps are not actually random, but they're coordinated in the right way, you can get it to spell out whatever you want. And that's exactly what language models are doing as a byproduct. There are a lot of statistical relationships in the structure of our data that they magnify and amplify in addition to all of the amazing things they do, like quasi Riemann. And understanding what those relationships are and how the models magnify and amplify them is really important to working with these systems, especially in context where we're deploying them and it's not just a very safe environment, like outputting a lean proof of theorem we care about, but it's about a messier problem that actually interacts with people.

是的，我的意思是，我们讨论过《生命游戏》和元胞自动机，从简单规则中涌现出令人难以置信的复杂性。同样地，你刚才谈论的是一个单一的大型 LLM 系统。如果你让那个系统进行大量推理，那已经是一个动态过程，可以与外部世界的工具交互。现在，你让多智能体系统或非常多的智能体群体系统。现在你有一种《生命游戏》元胞自动机，其中对猫头鹰的看似痴迷被发现，轻微痴迷于猫头鹰，导致大约 4 小时，比如说，随机选一个数字，突然所有人类都死了，到处都是猫头鹰。

Yeah, there's I mean, we've talked about Game of Life and cellular automata from simple rules emerges this incredible complexity. In that same way, what you've been talking about right now is a single large LLM system. If you take that one system and have it do a lot of reasoning, that is already this dynamical process that can interact with tools to the outside world. Now, you take that and have multi-agent systems or very multi-agent swarms of systems. And now you have a type of Game of Life cellular automata where the seeming obsession with owls that have been discovered, slight obsession with owls, results in about 4 hours, let's say, picking a number randomly, where all of a sudden all humans are dead and there's owls everywhere.

而这一切都始于随机数字。

And it all started from random numbers.

看，都只是随机数字。所以，这是一个漫画式的描述，但我们已经看到了证据。我道歉，有点把它带到了漫画化的地方，这使得它不严肃。我想退一步说，这些系统确实有很大的力量，因为我们在世界上如此依赖周围的数字系统。随着 AI 变得越来越强大，它对我们人类越来越有用，所以我们开始依赖它，给它更多访问权限，给它更多在公共世界中漫游的自由。现在，我们正在研究的这些行为，这些系统在规模上的数学，你现在开始得到元胞自动机效应，出现令人难以置信的复杂性，从简单的痴迷猫头鹰的智能体中产生的漩涡。所以，理解这种对猫头鹰的痴迷非常重要，这样它们就不会意外地对我们的文明造成大量损害，我们的文明在许多方面、许多层面上，在社会学上、地缘政治上、技术上都是脆弱的。我们存在的文明相当脆弱。因此，理解它们作为大规模快速连接的系统非常重要。所以，这非常真实，需要严谨，特别是可证明的严谨来理解其中一些系统。

See, all just random numbers. So, I it's a caricature, but we've already seen evidence of this. I apologize for kind of taking it so to the character place, which makes it not serious. I think to step back, these systems do have a lot of power because we depend so much in our world on our digital systems around us. And as the AI becomes more and more powerful, it becomes more and more useful to us humans, so we begin to depend on it, give it more access, give it more freedom to roam the public world. And now these behaviors that we're studying, the mathematics of these systems taken at scale, you now start to get the cellular automata effect of incredible complexity that emerges, the vortex that come out from simple owl-obsessed agents. So, understanding this obsession with owls is very important so that they don't accidentally do a lot of damage to our civilization that is in many ways, on many levels, and sociologically, geopolitically, technologically is fragile. It's a pretty fragile civilization that we're existing in. And so, understanding them as systems that connect at scale and very rapidly is very important. So, that's like that's a very real a to have rigor and especially like provable rigor to understanding some of these systems.

嗯，正是因为这个原因，我们也不想把如此多的决策外包给这些系统。

Well, we also don't want to outsource so much nearly so much decision-making to these systems for exactly that reason.

是的，我们已经为安全关键和高影响的决策建立了基础设施和协议。我们有 FAA，有美国医学协会，有律师协会来处理这些影响人类的事情。我认为，我们如何将 AI 融入全世界的通用流程，可以从两个轴来看。一个轴是我们对输出的信任程度，即可验证性或可信度。另一个轴是如果出错，风险有多高。如果 AI 输出一个错误的证明，没有人会断电，没关系，所以风险很低。而且它实际上非常可验证。所以你很快就能知道它对还是错，而且如果错了风险也很低。所以，完美的用例是开放式的设计空间。设计更好的车辆，更好的汽车设计，阻力更低，更省油。如果设计错了，风险很低，因为你总是可以在风洞中测试并验证它是否有效。所以，设计优化空间非常适合 AI。而我们人类必须非常小心，不要越过界限。我们需要一个防火墙，明确哪些需要实际验证。我们有协议，有系统，有法律体系和验证基础设施，但这是我们需要考虑的事情。如果我们变得懒惰，

Yeah, we have infrastructure and protocols in place already for safety-critical and high-impact decisions. We have the FAA, we have the American Medical Association, the Bar Association for these things that affect humans. And I think of how do we incorporate AI into our general processes in the entire world on two axes. We have how much do we trust the output? So, verifiability or trustworthiness. And on the other axis is how high-stakes is it if it's wrong. If AI spits out a wrong proof, nobody goes without power. It's fine. So, it's very low-stakes. And it is actually extremely verifiable. So, you know very quickly if it's right or wrong and it's low-stakes if it's wrong. So, that's the perfect use case is this open-ended design space. Designing a better vehicle, a better car design that's lower drag, more fuel-efficient. Low-stakes if it gets a wrong design because you can always test it in a wind tunnel and verify if it did or didn't work. So, like design optimization space is perfect for AI. And we as humans have to be very careful about not bleeding into We need a firewall of what needs to actually be verified. And we have protocols, we have systems, legal systems and infrastructure for verification, but it is something we need to think about. If we get lazy,

是的。

Yes.

我们可以 这是一个非常滑的斜坡。我们需要有那个防火墙。

we can It's a very slippery slope. We need to have that firewall.

嗯，你认为目前在数学领域，或者对你来说，史蒂夫，也许在更广泛的数学之外的空间，摆在我们面前的最难的问题是什么？你认为哪个问题不太可能在接下来一两年内解决？

Um what do you think are the hardest problems before us right now in the space of mathematics, or in your um maybe for Steve more in the space a little bit broader outside of mathematics. What do you think is the problem that's unlikely to be solved in the next year or two?

我仍然认为是 P 与 NP 问题。

I'm still going with P versus NP.

你认为 顺便问一下，你认为有没有可能最终发现 P 等于 NP？

You think that By the way, do you think it's possible it turns out that P equals NP?

当然，一切皆有可能。

Sure, anything's possible.

那个世界在你看来是什么样子？

What does that world look like to you?

所以，这取决于算法的运行时间。

So, it depends on what the running time of the algorithm is.

对。

Right.

所以，它可能大到不切实际，那么就不那么重要了。这对理论研究更像是一场海啸。是的，但那似乎是一个非常重要的问题，而且很难解决。事实上，我记得当我还是研究生时，有一个关于 P 与 NP 的尝试，实际上是半认真的。有些人开始传言说：“嘿，它被解决了吗？”我的第一反应是：“不，它没有被解决。”我记得花了一个下午思考，当时博彩市场还相当新。所以，它不允许我对“P 与 NP 尚未解决”进行反向投注。所以，相反，我投注了所有其他千禧年大奖难题，因为那似乎是一个很好的对冲方式，因为问题是哪个千禧年大奖难题会首先被解决。

So, it could be so large that it's impractical, and then it doesn't really matter that much. It's more of a tidal wave for the theory research. Yeah, but that seems like a very important problem and very hard to fall. In fact, I remember back when I was a graduate student, there was an attempt on P versus NP, which was actually semi-serious. Some people started all having rumors that, "Hey, has it been solved?" And my first reaction was, "No, it has not been solved." And I remember spending an afternoon thinking about the betting markets were still pretty new then. So, it wouldn't let me take a counter bet to P versus NP has not been solved. So, instead what I did was I bet on all the other Millennium Prize problems, cuz that seemed like a good way to hedge, because the question was which Millennium Prize problem will be solved first.

嗯。但是，有没有可能 P 等于 NP 是以一种真实的方式？也就是说，我们对复杂性理论、计算机科学、算法有一些深刻的不理解，以至于存在用于可满足性的快速算法？

Mhm. But, uh is it possible that P equals NP in the real way? So, there's something deeply that we don't understand about complexity theory, about computer science, about algorithms, where there are fast algorithms for satisfiability?

所以，我认为这里面有两条线索需要梳理。首先，至少到目前为止，似乎你总是可以制造出问题的困难实例。我们利用这一点在密码系统中取得了巨大效果，但在实践中，很多问题似乎比我们认为的要容易得多。所以，对于 SAT 之类的问题，它们出现的地方之一是编译器和寄存器分配。

So, I think there are two threads in there that are important to tease out. So, first, it seems, at least so far, that you can always manufacture hard instances of problems. And we use that to great effect for our cryptography systems, but it does seem like in practice a lot of problems are much easier than we give them credit for. So, for things like SAT, one of the places where they come up is in compilers and register allocation.

SAT 求解器在解决这类 SAT 问题上效果惊人。我们并不真正清楚，与来自密码学或其他领域的其他难题相比，是什么让这些问题如此容易解决。但可能在实际中，我们建模的许多最坏情况问题其实是容易的。而更难的部分是阐明它们具有什么样的结构，使它们变得容易。但仍然有丰富的难题来源，我们可以做很多事情，比如构建密码系统。这是我对我们所处世界的赌注。但这只是猜测。

And SAT solvers are amazingly effective at solving those kinds of SAT problems. Now, we don't really know what makes those problems so eminently solvable compared to the other hard problems that come out of cryptography or other areas. But it could be that in practice a lot of the problems that we model in the worst case are secretly easy. And that the harder part is articulating what kinds of structures they have that makes them eminently easy. But that there are still a rich source of hard problems that we can do a lot of things, like building cryptographic systems out of. That would be my bet in the world we live in. But that's just a guess.

所以，也许有某种方法可以快速分类出这个特定问题实例是容易的。也就是说，能够快速识别出这是较容易的实例之一。就像有一个子集。

So, maybe there's some way to basically quickly classify that this particular example of the problem is easy. So, like be able to identify quickly that this is one of the easier instances. Like there's a whole subset.

有可能。这让我想起《安娜·卡列尼娜》的开场白：幸福的家庭都是相似的，不幸的家庭各有各的不幸。我的担忧是，许多问题比我们想象的要容易，

It could be. So, I'm reminded of the opening line for Anna Karenina: every happy family is happy in the same way, and every unhappy family is unhappy in their own way. I think my worry is that many problems are easier than we make them out to be,

是的。

Yeah.

但它们容易的原因各不相同。这并不意味着算法视角在分类或开发这些结构的本体论方面没有用，但我认为来自寄存器分配的 SAT 问题类型与来自其他领域的 SAT 问题非常不同，尽管它们可能容易。我只是认为它们容易的原因非常不同。

but they're easy for very different reasons. Now, that doesn't mean that algorithmic perspectives aren't useful in trying to classify or developing an ontology of what those structures are, but I think the types of SAT problems that come out of register allocation are very different than the other SAT problems that come out of other domains, even though they might be easy. I just think they're easy for very different reasons.

所以，对你来说，难题是 P 和 NP。所以，如果在接下来几年内解决了，你会印象深刻？

So, to you, the hard problems are P and NP. So, you'll be impressed if within the next couple of years that's solved?

我会相当印象深刻。

I would be pretty impressed.

Alex，什么难题会让你印象深刻？

Alex, what's the hard problem that you'll be impressed by?

嗯，我已经对 quasi-RH 印象深刻了，但它不是 RH。你知道，很难想到一个更具标志性的问题来说这是最难的，数学中最难的问题之一。它已经存在了这么久，而且

Well, I'm already impressed with quasi-RH, but it's not RH. You know, it's hard to think of a more iconic problem to say this is the hardest, you know, one of the hardest problems in mathematics. It's been around for so long and

你对此怎么看？抱歉，这让我想起了庞加莱，比如佩雷尔曼。他说了什么？他说如果证明是正确的，那就是我需要的所有荣誉。他著名地拒绝了菲尔兹奖。你认为他会对 OpenAI 的这次发布说些什么？

What do you think on that? Sorry, it just made me think of Poincaré like the Perelman. What did he say? He said if the proof is correct, that's all the credit I need. He famously turned down the Fields Medal. What do you think he would say about all this on the OpenAI drop?

我的意思是，我很难预测佩雷尔曼会做什么。我不知道他会拒绝菲尔兹奖。我不知道他会拒绝百万美元的克莱奖。

I mean, I have a very hard time predicting what Perelman will do. I did not know he would decline the Fields Medal. I did not know he would decline the million-dollar Clay Prize.

你对此总体上怎么看？比如拒绝奖项。在奖项历史上，有几个人这样做过。他是唯一一个拒绝克莱奖的人。

What do you think about that in general? Like turning down a prize. A few folks have done that in history of prizes. He's the only one who has turned down a Clay Prize.

是的，还有一百万美元。呃，我怎么看？数学家很古怪。

Yeah, and the million dollars. Uh what do I think about it? Mathematicians are weird.

好吧。

Okay.

古怪。各有所好。我们热爱观点和思想的多样性，这正是数学世界如此美妙的原因，有那么多不同种类的人，我们让他们都加入进来。

Weird. To each his own. We love diversity of opinions and thoughts and that's what makes the world of mathematics so wonderful is that there's so many different kinds of people and we let them all in.

我的意思是，这些问题与数学界现在正在处理的事情以及奖项的价值有关，因为在某种意义上，他特定的世界观与数学界正在进入的新世界观是一致的。这甚至说不通。菲尔兹奖——我不知道是否还会有另一个菲尔兹奖颁发。所以你会说我们正在进入问题世界。

I mean, these questions are kind of coupled with what the mathematics community's dealing with now and the value of prizes because in some sense what his particular worldview is aligned with this new worldview that the mathematics community's entering. Which is like it doesn't even make sense. The Fields—I don't know if there'll be ever another Fields Medal given. And so you're like we're entering problem world.

今年夏天我们明确地讨论了这个问题，当时菲尔兹奖，你知道，上一轮菲尔兹奖在费城颁发，确实有一个问题，因为人们一直在关注进展。还没有这最新的进展，真正以最有力的方式提出了这个问题，

We were talking about that explicitly this summer when the Fields Medals, you know, the last round of Fields Medals was given out in Philadelphia and there was definitely a question, you know, cuz people were following the progress. There hadn't been this latest bit of progress that really puts the question in the most powerful way possible,

可能的，但是

possible, but

嗯，今年夏天我们已经在想，这是最后一次颁发菲尔兹奖吗？我当然希望不是。但我们必须弄清楚这意味着什么，你知道，你必须更社会学地思考奖项能实现什么？为什么要有这些奖项？获得这些奖项的人已经是顶尖地方的终身教授，他们将在余生继续做数学。嗯，佩雷尔曼是一个

um already this summer we were thinking is this the last time that a Fields Medal is awarded? I sure hope not. Uh but we have to figure out what that means, you know, you have to think more sociologically about what do prizes accomplish? Why have these prizes? The people who win these prizes are already, you know, tenured professors at top places and they're going to be doing mathematics for the rest of their lives. Well, Perelman being one

呃，一个非常

uh a very

是的，例外。

Yeah, exception.

非常呃著名的例外。

very uh well-known exception.

那么奖项能实现什么？它们庆祝这项活动。它们庆祝。它们说这是我们认为重要的。这是某事的最高实例。仍然会有我们想要庆祝的活动。我不知道现在它们是否因为解决重大问题而颁发。解决重大问题是 AI 系统能做到的事情。这不会稀有，因此它不会像过去那样有价值。我想认为，尽管我们之前讨论了已解决公理，再次，我不相信已解决公理，我相信半人马数学将是一个非常

So what do the prizes accomplish? They celebrate the activity. They celebrate. They say this is what we think is important. This is, you know, the highest instance of something. And there will still be activities that we want to celebrate. I don't know if right now they're given out for solving major problems. Uh solving major problems is something AI systems can do. That's not something that will be rare and hence it won't be nearly as valuable as it was in the past. I would like to think that despite our earlier discussion about the solved axiom, again, I don't believe the solved axiom and I believe centaur math will be a very

半人马数学，好的。

Centaur math, okay.

比如做数学的好方法。我的意思是，这就是我现在的感觉。我经常与 AI 系统交谈，我学习数学的速度比我能做到的快得多，因为数学的写作方式，你知道，我有时给研究生一篇论文读，他们一周后回来，我说：“那么，你读到哪了？”“嗯，说实话，我没读过第一页。”“哦，真的吗？告诉我。”“嗯，实际上，我没读过第一段。”我说：“好吧，为什么没有？”“嗯，第一句话是‘非局部系统的微扰理论’，我不知道那是什么意思，所以我卡在那里了。”我说：“好吧，你不知道如何读数学论文。你跳过前几段。直接到引理三，从那里开始工作，然后回溯到它在做什么。”但是，在未来，没有“我不知道这是什么意思”，因为你只需上传到 LLM，然后开始问它问题，它就会聚焦在最近发展区，即这里是你知道的一切，这里是你需要听到的下一件事，以便你开始学习。如果人们有效地使用它，他们真的可以超级充电他们的教育。我不仅仅指孩子，我指像我这样的老家伙可以学习新领域，你知道，学习一个领域。就像，这就是我们的研究生正在经历的。他们必须完全重新连接他们的大脑。就像，学习数学仍然很难。

like great way to do math. I mean, that's how I currently feel now. I'm constantly talking to AI systems and I'm learning a lot of math a lot faster than I'd be able to because the way math is written, you know, I sometimes give a paper to a grad student to read and they come back a week later and I say, "So, how far did you get?" "Well, I didn't really get past the first page, to be honest." "Oh, really? Tell me." "Well, actually, I didn't get past the first paragraph." I said, "Okay, fine. Why not?" "Well, the first sentence is by the perturbation theory of non-local systems and I don't know what any of that means, so that's kind of where I'm stuck." I'm like, "Okay, you don't know how to read a math paper. You skip the first couple of paragraphs. Go straight to lemma three and that's where you can start working and work your way backwards to what the thing is doing." But, in the future, there is no "I don't know what this means" because you just upload it to an LLM and you start asking it questions and it will zero in on the zone of proximal development, the place where here's everything that you know and here's the next thing you need to hear so that you start learning. And if people use that effectively, they can really supercharge their education. I don't just mean kids, I mean old fogies like me can learn new fields, you know, to learn a field. Like, this is what our graduate students are going through. They have to completely rewire their brains. Like, learning math is still hard.

嗯。

Mhm.

直到硕士水平进入博士水平，你才被期望产生新的数学理论。他们能做的一切都是教科书里有的东西。

Up until past the master's level into the PhD level, you were not expected to produce new mathematical theories. Everything that they can do is something that is in a textbook somewhere.

然而，对人们来说，学习解决那些许多人都能解决的问题仍然极其困难，更不用说那些尚未解决的问题了。所以，学习数学很难，学习新的数学更难。发现数学或找出数学问题的解决方案……但我认为这会变得更容易，这个工具将成为一个惊人的资源，让人们能够更快地进入最近发展区，并更快地发展他们的理解。

And yet, it's still extraordinarily difficult for people to learn to solve problems that are solvable by lots of other people. Never mind problems that aren't solved. So, learning math is hard. Learning new math is even harder. Discovering math or working out solutions to math problems... But I think it's going to get easier and this tool is going to be an amazing resource for people to be able to get to that zone of proximal development much more rapidly and develop their understanding much more rapidly.

是的，我一直在一些我甚至不是专家的主题上玩得很开心，我很想带着我现在拥有的工具回到研究生院。但我的大脑喜欢学习的方式之一是构建视觉示例、玩具示例，即仍然包含洞察力的最简单可能的例子，特别是如果它是交互式的，就像如果我能移动东西。就像，哦，你立刻就能理解。

Yeah, I've been having so much fun on topics I'm not even [expert in], and I would love to go back to grad school with the tools that I have now. But one of the ways my brain likes to learn is constructing visual examples, toy examples, the simplest possible example that still contains the insight there, and especially if it's interactive, like if I can move stuff. It's like, oh, you immediately get it.

有所有这些社区约定俗成的词汇，你应该知道所有的符号，所有这类东西。如果我能知道这个小符号代表这个，哦，好的，酷。那么一切就都迎刃而解了。而且，你会爱上这个主题，我觉得我可以解决任何完全超出我领域的问题。比如我可以理解蛋白质折叠。我可以摆弄参数玩。我可以拿一篇 DeepMind 的论文，通过选取它的任何方面并玩弄它来真正理解，什么是简单的例子，问一些疯狂的前面提到的瘾君子问题。所以，是的，这超级有趣。问题是人们是否有动力去学习。这是问题所在。这是人类动机的问题。那才是真正的问题。

There's all these words that the community agreed means something, you're supposed to know all the notation, all that kind of stuff. If I can just know that this little symbol means this, oh okay, cool. All right, that and it just everything kind of falls into place. Also, you fall in love with the subject and I feel like I can take on any problem that's completely outside of my field. Like I can understand protein folding. I can play with the parameters and play. I can take a DeepMind paper and actually understand by taking any aspects of it and playing with it and what's the simple example, ask crazy aforementioned pothead questions. So, yeah, this is super fun. The question is whether people will be motivated to learn. This is the problem. It's the problem of human motivation. That's the real question.

嗯，关键要素是好奇心、提出好问题，以及能够调用批判性顾问技能，比如在你不是专家的领域获取信息，并知道是否需要深入挖掘或反驳。我认为，你知道，总会有好奇和不那么好奇的人，以及会问好问题和不会问的人。我们需要比以往任何时候都更多地推广这一点。让我有点害怕的是，我们期望我们的本科生、年轻学生在学习时与这些工具互动，而他们的大脑尚未完全发育。我们要求他们具备批判性，不仅仅是批判性推理能力本身，还有顾问技能。你如何吸收信息并判断它是否真实，或如何质疑它。这对我们教育来说是一个巨大的任务，即我们如何

Well, the key ingredients there are curiosity, asking good questions, and being able to call like the critical advisor skills of like getting information where you're not the expert and knowing if you need to dig deeper or push back. And I think, you know, there will always be people who are curious and not curious and who ask good questions and don't. We need to be really promoting that more than we ever have. It's a little scary to me that we're expecting our undergraduates, our young students, to interact with these tools when they're learning, when their brains aren't fully formed. We're asking them to have critical, not just critical reasoning skills themselves, but like advising skills. How do you take in information and tell if it's true or not, or how to question it. And that's a huge task for us in education is how do we

是的。

Yeah.

在孩子们小时候训练他们拥有良好的顾问技能。

train people to have good advisor skills when they're kids.

是的，我认为最难的问题不是本科。最难的问题像是中学。因为就像那个，作业开始变难的临界点是什么？

Yeah, I think the hardest problem isn't undergrad. The hardest problem is like middle school. Cuz like that I What is the point at which homework starts getting hard?

是的。

Yeah.

在那个时候，为什么要做作业？

At that point why do the homework?

因为诱惑，对吧？

Cuz the temptation, right?

那东西就会……起初，我说，看，由于所有的痛苦，我已经体会到做困难事情的价值，但我认为那是因为我被强迫的。但如果你不是被强迫去做，因为他们都能……我可以帮你。我不知道那会创造什么样的社会。

That thing will just At first, I'm saying See, I've come to appreciate the value of doing difficult things because of all the suffering, but I think because I was forced to. But if you're not forced to do it because all of them can they I can just help you. I don't know what kind of society that creates.

但数学在那个问题上有着最长的先发优势，因为计算器是什么时候出现的，70 年前？我们仍然要求孩子们学习乘法表，至少我们中的一些人这样做。而那些说“不，孩子们不需要学习乘法表”的人，你知道，到老师和孩子们听信了这种说法并且没有学习的地步，他们真的受苦了。大量的人类苦难源于人们说“你不需要知道乘法表，因为计算器可以做”。我希望我们能从中吸取教训，说“仅仅因为计算器可以做，如果你自己也能做，你会过得更好”。这并不意味着如果你……如果我必须做一些计算，显然我会伸手去拿计算器。但我原则上可以自己做，这就是为什么每次你给自己那个技能并掌握一个技能。孩子们喜欢掌握技能。他们第一次学会系鞋带时有多高兴？你会想这就像只是鞋带，谁在乎？不，这是最伟大的事情。你想庆祝每一点独立，你想能够思考。你想独立于机器。仅仅因为机器能做某事并不意味着你不想拥有自己做的能力。

But math has the longest head start on that question because calculators came out what, 70 years ago? And we still ask kids to learn their times tables, at least some of us do. And those who said no, kids don't need to learn times tables, you know, to the extent that teachers listened to that and kids listened to that and didn't learn, they really suffered. There's a great amount of human suffering resulting from people saying you don't need to know times tables cuz calculators can do it. And my hope is that we can learn from that and say just because a calculator can do it, you're going to be much better off if you can also do it yourself. Doesn't mean you if I have to do some calculation, obviously I'm reaching for a calculator. But I could in principle do it myself and that's what allows every time you give yourself that skill and master a skill. Kids love mastering skills. How happy are they when they learn to tie their shoelaces for the first time? You would think this like it's just shoelaces, who cares? No, it's the greatest thing ever. You want to celebrate every little bit of independence and you want to be able to think. You want to be independent from the machines. Just cuz the machine can do something doesn't mean you don't want to have the capacity to do it yourself.

嗯，这就是我必须问的地方，有很多人充满焦虑。我们提到了研究生、博士生、本科生、高中生，你会给什么建议？也许对不同层次的人有不同的建议。嗯，对那些试图弄清楚自己人生该做什么的人，对现在惊慌失措的博士生。你能给他们一些智慧吗？

Well, this is where I have to ask there's a lot of people that are full of anxiety. We mentioned graduate students, PhD, graduate, undergrad, high school students, what advice would you give? Maybe to different advice at different levels. Um to folks trying to figure out what to do with their life, to PhD students who are freaking out right now. Is there some wisdom you can give them?

所以，是的，每个尺度都有不同的东西。让我从博士生开始，我认为在数学方面，纯数学的博士生，我给他们的建议非常简单。做我们一直做的事情。学习你的头脑能处理的最困难、最抽象的数学，尽可能多。你的导师应该希望引导你完成这个过程，然后我再往下走一步。我们在本科层面看到的是双峰分布。过去成绩基本上是高斯分布。现在我们看到非常双峰的分布，在那些使用 AI 检查理解的孩子之间。“等等，我没懂。教授在课上说了这个。那是什么意思？我有个考试要来了。这是我过去 3 周的笔记。开始测验我，检查我是否真的知道。”那些孩子得到更高的分数。然后有些孩子，“哦，我这个习题集一小时后要交。你能直接告诉我答案吗？我会用自己的话写下来。”所以，作业进度和考试之间出现了脱节。我们看到了这种双峰分布。我不知道你们是否看到同样的情况。所以，隐含的建议是成为那些使用 AI 来学习和成长的人，成为分布的头部而不是尾部。

So, yeah, there's different things at every scale. Let me start with PhD students, I think in math, PhD students in pure mathematics, my advice to them is very simple. Do what we have always done. Learn the most difficult, the most abstract mathematics your mind can handle, as much of it as possible. Your advisor should hopefully be guiding you through that process and then just I'll go one more one step down from that. What we're seeing at the undergraduate level is a bimodal distribution. It used to be grades were basically Gaussian distributed. Now we're seeing a very bimodal distribution between the kids who are using AI to check their understanding. Wait, I didn't get it. Professor said this in class. What did that mean? I have a test coming up. Here are my notes from the last 3 weeks. Start quizzing me and check if I really know something. Those kids are getting much higher grades. And then there are kids who oh, I this problem set is due in an hour. Can you just tell me the answers? I'll write them down in my own words. And so, there's like a decorrelation between homework progress and exams. And we're seeing this bimodal distribution. I don't know if guys are seeing the same thing. So, the implied advice there is to be one of those the one that's using AI to learn and to grow to be the head not the tail of the distribution.

是的。你介意我们休息一下去洗手间吗？

Yeah. Do you mind if we take a bathroom break?

好的，好的，好的。

Yes, yes, yes.

我们不知怎么地把智力变成了这个巅峰，几乎把智力等同于道德。

We've somehow made intelligence this pinnacle and have almost equated intelligence with morality.

对。或者价值。

Right. Or value.

价值。我认为这再错误不过了，对吧？就像超级智能并不意味着超级道德。

value. And I think that couldn't be farther from the truth, right? Like superintelligence does not imply super morality.

我认为这是 AI 系统的一个巨大隐患：即便它们超级智能，也并不意味着它们会超级有道德。

And I think that's a huge concern with AI systems: if they are superintelligent, it has no implication that they will be super moral.

对。

Right.

而且我觉得这么多年来我们一直推动孩子上大学，好像每个人都需要去。

And I think we've been pushing kids to go to college for however many years, that everybody needs to.

不管他们学什么，或者结果如何。

Regardless of what they study or what the outcomes are.

没错。

Exactly.

而超级智能似乎赋予你一种超级合理化能力。有常识的人，比如世界各地的农民，他们有一种感觉：合理化往往只是因为你能够把它说得天花乱坠，而我无法反驳，但这并不代表它是对的。

And superintelligence seems to give you a super ability to rationalize. And common sense people, you know, farmers in the rest of the world, they have a sense that that rationalization is often just because you can make it sound really good and I can't argue with it doesn't make it right.

对。教育和智慧是两回事。

Right. Education and wisdom are two different things.

是的，没错。我知道这很难做到。

Yeah, exactly. I know it's difficult to do.

但也许 Steve，你对研究生、大学生，或者年轻人有什么建议吗？在这个疯狂快速变化的时代该怎么做？

But maybe Steve, do you have advice for grad college, maybe young kids, on what to do in this crazy fast-changing time?

一定要保持好奇心，阅读并学习所有基础知识。我认为这始终是很好的建议。即使你的电脑能比你更好地做线性代数，你仍然应该学习线性代数，建立直觉。微积分、物理也是如此。我认为理解历史，并在人类存在过程中发生的其他巨大变化的背景下理解这次变革，是很重要的。但真正重要的是保持好奇心。

Definitely be curious and read and learn all of the fundamentals. I think that's always been good advice. Even if your computer can do linear algebra better than you, you should definitely still learn linear algebra, build the intuition. Same with calculus, same with physics. I think understanding history and understanding this change in the context of other great changes that have happened through human existence is important. But really being curious.

我得说，头脑中的这种好奇心正不断被参与经济所窃取。所以，让你好奇的同一引擎，也是让你无休止刷屏的引擎。

I should say that this curious thing in the mind is trying to get stolen non-stop by the engagement economy. So like the same engine that makes you curious is the same engine that makes you doomscroll.

就像，哦，那是一个——

Like, oh, that's a—

所以我认为你必须学会如何避免所有干扰，专注于让好奇心真正探索事物并吸收一些东西，比如学习一样东西。实际上 AI 非常擅长这样做，因为它们被激励从互动中为你提供价值，而不是被激励让你粘着、上瘾。

So I think you have to kind of learn how to avoid all the distractions and focus on letting the curiosity actually explore things and soak something in, like learn a thing. And actually AI is really good at doing that because they're incentivized to give you value from the interaction versus incentivized to get you glued, addicted to the thing.

嗯。

Mhm.

我不知道高中、大学的人怎么能在手机就在身边的情况下生存。所以干扰，那个干扰盒子就在那里。所以避免干扰，深入专注。

I don't know how people in like high school, college also can just survive with the phone being around. So the distraction, the distraction box being right there. So avoid distractions and focus deeply.

我们小时候有时都会无聊。我认为无聊真的很重要，真正地无聊，就像好奇和专注往往来自无聊。我希望自己能更经常无聊。我有时会尝试这样做。

We all were bored when we were kids sometimes. I think boredom is really important to actually be bored, be like curiosity and focus comes often from boredom. I would like to be bored more often. I try to do that sometimes.

是的，我们放学回家，你知道，5 秒钟做完作业什么的，然后就出门，你希望碰到其他也刚出门的孩子，然后你去找点事情做，晚饭时间才回家，而且

Yeah, we would come home from school and like, you know, finish homework in 5 seconds or whatever and just go outside and you hope to run into some other kids who also just went outside and then you like go and find something to do and come home when it's dinner time and

是的。

Yep.

是的，你实际上没有意识到，就像高中几年，甚至某种程度上大学之前，但开始变得越来越难。但高中是一个非常好的时期，可以学习一切。你想，至少我记得，我不知道现在是什么样子，但我觉得现在甚至更有趣。你想，好吧，当我长大了，我就能在这个世界上做点有用的事情。我，是的，我明白学习很好等等。但你意识到，从实现满足感的角度，从构建美好生活的基础、让你记住所有这些事情的角度，这就是该做的时候。整个世界都在强迫你这样做。他们会奖励你坐在那里学习。因为当你更深入，你知道，甚至到研究生院，这种探索性的事情，你会受到阻碍，天哪，如果你找到工作，那你肯定会被阻碍去学习、探索等等。所以，探索的时间就像高中。顺便说一句，就像

Yeah, you don't realize actually like when the high school years even before somewhat undergrad but starts getting harder and harder. But like high school is a really nice time to just like learn everything. And you think like I at least I remember, I don't know what it is like now, but I think it's just more even more fun now. You think you like, okay, when I grow up, I'll get to actually do something useful in this world. I yeah, I get I'm learning is very nice and so on. But you realize like this is like from a fulfilling perspective, from building foundation of an awesome life where you remember all of these things, this is like the time to do it. And the whole world is forcing you to do it. And they will reward you for sitting there and learning. Cuz like when you get more, you know, even to like grad school, this kind of exploratory stuff, you get discouraged and uh God forbid you get a job, then you're definitely discouraged from just uh learning, exploring, and so on. So, the time to explore is like high school. By the way, like

终身教职前的教师就是这样，哦，一旦我拿到终身教职，我就可以坐下来放松，思考大问题。但我首先得产出足够的东西来获得终身教职。某种程度上，我们不希望我们的年轻人做的就是拼命爬上去。

Pre-tenure faculty are on this like, oh, once I get tenure, then I can just sit back and relax and think big thoughts. But I have to like produce enough stuff to get the tenure in the first place. And sort of the it's not what we want our young people to be doing is clawing their way there.

Uncle，你对学生有什么建议吗？

Uncle, do you have advice you give to students?

我最后讲，我觉得我的很多建议已经被说了。所以，保持好奇心肯定是其中之一。现在，我想说，为内在原因做事也很重要。我注意到的一件事是，我们行业中很多人，很多走学术路线的人，他们做事的原因有时真的是外部自我定位，因为他们通过别人如何看待自己来看到自己的价值。这足以让你起步。也许你开始非常努力地学数学，以便让人们对你印象深刻，对吧？但到了某个时候，你用完了这些里程碑，然后你真的必须有内在原因来做你做的事情。希望你有意识，知道什么对你重要，你的价值观是什么。所以，我认为现在，你可以采纳她关于好奇心的建议，但旧的剧本真的被抛弃了。所以，你可以采纳她的高层次建议，但你必须自己弄清楚，你必须对不确定性感到舒适。而我知道处理这种不确定性的唯一方法是，对你认为重要的事情、你热爱的工作有一种使命感，并确保你正在做这些事情。因为旧剧本，如果你只是坚持它，你最终会接受所有这些 AI 生成的证明，并花所有时间把它们放到 arXiv 上，发送给期刊。我不认为任何人进入这个领域的内在原因真的是为了充实简历。我不知道未来什么会重要，但如果你真的有一个你相信的更高目标，那么我认为你将处于更有利的位置来适应变化的世界，制定自己的蓝图，并在周围世界变化时对不确定性感到舒适。

With going last is that I think a lot of my advice got taken. So, be curious was definitely one of them. Now, I would say that it's also important to do things for intrinsic reasons. So, one of the things that I've noticed is, you know, a lot of people in our profession, a lot of people who go the academic route, the reason they do things sometimes is really external locus of self because they see their value through how other people see them. And that's good enough to get you started. Maybe you start working very hard math so that people are impressed by you, right? But at some point you run out of these milestones, you know, to hit, and then you really have to have an intrinsic reason for doing the things you do. And hopefully you have a sense of, you know, what's important to you, what are your values. So, I think these days, you know, you can take her advice about being curious, but also the old playbook is really being thrown out. So, you can take her high-level advice, but you're going to have to figure it out for yourself, and you're going to have to be comfortable with uncertainty. And the only way I know how to deal with that kind of uncertainty is to have a sense of purpose about what you think is important, what you love about your job, and make sure that those are the things that you're doing. Cuz the old playbook, if you just stick to that, you're going to end up taking all these AI-generated proofs and spending all your time putting them on arXiv, sending them to journals. I don't think that's anyone's intrinsic reason if they get into it is really to pad their CV. I don't know what'll be important in the future, but if you really have some higher purpose you believe in, then I think you'll be in a much better position to adapt to the changing world and figure out your own blueprint and be comfortable with uncertainty as the world changes around you.

这是一个更棘手的问题，但很多听众对未来的感到很多焦虑和恐惧。这既关乎工作，也关乎一个有意义的未来。你能回顾一下自己的生活，当你遇到那种不确定性、那种焦虑时？你能给那个家伙什么建议？当你处于低谷时。

This is a trickier question, but a lot of people listening to this are feeling a lot of anxiety and fear about the future. But it's both about a job, but also a future that's meaningful. Can you look back to your own life when you had that uncertainty, that anxiety? What's the advice you can give to that guy? That to when you're in that low point.

那么，让我试着回答这个问题。这是一个非常难的问题。我讲一个短故事，你知道，我们在湾区度学术休假，住在帕洛阿尔托，那里算是大学压力的中心。你知道，你能感觉到。你去咖啡馆，那时压力就在空气中。你知道，你现在在做什么课外活动会对你有帮助？然后就在我回到 MIT 时，他们让我在那里做毕业典礼演讲。

So, let me try to address that question. It's a very tough question. I'll tell a short story that, you know, we spent our sabbatical in the Bay Area, and we're living in Palo Alto, which was kind of the epicenter of college stress. You know, you could just feel it. You go to the cafes, it's just in the water at that point. You know, what extracurriculars are you doing right now that will help you? And then right when I came back to MIT, they asked me to give the convocation speech there.

想想那场演讲就很有意思，你知道，面对整个礼堂的新生和他们的家人，你会说什么？如果一切都是一场游戏，目的是优化你作为一个 17 岁孩子在纸面上的表现，那他们已经赢了。我一直在想，当我处于同样的情况，刚开始上大学时，我是多么兴奋，仅仅是为了学习所有这些不同的东西。你知道，我会坐在那里规划我的四年，我有小笔记本，用来规划我要上哪些课，如果我按正确的顺序选这些课，我就可以在大二修光子学。我根本不知道光子学是什么，但它听起来很有趣、很高级。我觉得这方面，你知道，我们社会现在太多东西都朝着赢和在纸面上好看的方向发展。我总是想知道这到底是为了什么。你从高中开始，一进门就可以开始优化你的 AP 课程和课外活动，以便进入精英学校。然后当你进入那所精英学校，你可能会逃课，因为也许重要的是准备面试，以便在前沿实验室找到工作。然后你可以得到一些不错的限制性股票单位，然后过完你的一生。那很好。你赢了。你在每个阶段都赢了，但那又怎样？你知道，对我来说，过上丰富而充实生活的最好方式就是假装你已经赢了，然后弄清楚在那种情况下你会做什么。我记得我开始在 MIT 任教时得到的建议，Sanjiv Arora 告诉我，获得终身教职最简单的方法就是假装你已经拥有了它，然后做你一直想做的事。

And it was just interesting thinking through that speech, you know, what do you say to this entire auditorium filled with freshmen and their families who if things are a game to optimize how good you look on paper as a 17-year-old kid, they've won. And I just kept thinking about when I was in that same situation, when I was starting college, how excited I was just to learn all of these different things. You know, I'd sit there and plan out, uh, you know, my four years and you know, I had little, uh, notebooks where I'd figure out what my classes would be and if I took these classes in just the right order, I could take photonics as a sophomore. And I had no idea what photonics was, but it sounded fun and fancy. And I think that aspect, you know, so much of our society now is geared towards winning and looking good on paper. And I always wonder to what end. You start off in high school and you can start as soon as you walk in the door optimizing your APs and your extracurriculars so that you can get to an elite school. And then when you get to that elite school, you might skip classes because, you know, maybe the important thing is to prepare for your interviews so that you can get a job at a Frontier Lab. And then you can get some nice, you know, restricted stock units and and live out your life. And that's great. You won. You won at every stage, but so what? You know, for me the, you know, the best way to live a rich and fulfilling life is just to pretend you've already won and then figure out what you would do in that scenario. I remember the advice I got when I started at MIT as faculty, Sanjiv Arora told me, you know, the easiest way to get tenure is just to pretend you already have it and do what you would have wanted to do all along.

正是如此。

Exactly.

所以，我不知道这个建议是否也适用于研究生、高中生，但我想我的建议是一样的。假装你已经获得了终身教职。你现在会做什么？因为我们给出建议的很多人确实有太多不同的机会。找到你真正相信的东西是我所知道的获得满足感的唯一途径。

So, I don't know if that advice applies to, you know, graduate students, to high schoolers, too, but I think my advice stays the same. Pretend you already have tenure. What would you now do? Cuz a lot of the people we're giving advice to have really so many different opportunities. Finding something that you really believe in is the only way I know to be fulfilled.

我从一位我尊敬的人那里听过一条职业建议：一个充实的职业是你能解决一个重要的问题，你觉得它有趣，而且你擅长它。如果你能得到这三者中的两个，你就很幸运了。如果你能得到全部三个，你知道，你余生都会快乐。所以，做好正确的准备，弄清楚你擅长什么、什么有趣、什么重要，然后准备好自己，你不能保证得到全部三个，但你要在机会来临时做好准备。然后，继续做你真正喜欢做的事情。我觉得这是很好的建议。

I've heard uh career advice uh from someone I respected that a fulfilling career is something where you can work on a problem that is important, that you find interesting, and that you're good at. And if you can get two out of those three, you're pretty lucky. If you can get all three, you're going to be, you know, happy for the rest of your life. And so, having the right preparation, figuring out what you're good at, what's interesting, what's important, and then preparing yourself, you're not guaranteed to get all three of those, but you want to be ready when that opportunity comes. And then, continue doing the things you actually enjoy doing. I think that's great advice.

我觉得我们也应该结合背景，并且相当诚实。我确实认为你给出的很多建议都适用，但我们应该诚实地说，我面前坐着三个世界级的大脑。嗯，并不是说他们已经找到了自己的道路。你这是幸存者偏差。所以，你知道，有很多人在听这个节目时会想：他们很确定自己找不到自己的道路。这是恐惧，是焦虑，是不确定性。是跌跌撞撞了很多次。我觉得公平地说，一些最有趣的人，对吧，在生活中也会跌跌撞撞。无论是外部环境还是内部因素，或者宇宙带来的任何东西。在这方面，我知道很多人现在只是感受和观察着这一切，因为 AI 对我们社会所做的一切，很多人只是感到恐惧和非常焦虑。我觉得 Ankur 说的很有智慧。你可以被这种恐惧麻痹，或者你可以想象我已经赢了。那个已经赢了的人在这种情况下会怎么做？这减轻了恐惧，它实际上给了你在不确定性下的最优路径。在高度不确定性下，在很多不确定性下。不要用旧剧本，试着成为那个想出如何写新剧本、下一个剧本的人。

I think we should also put into context and be quite honest. I do think a lot of the advice you gave applies, but we should be honest that I'm sitting with three world-class human brains. Um not that have figured found their path. You're survivor bias. So, you know, there's a lot of people listening to this that are like It's like they're pretty sure they won't find their path. It's the fear, it's the anxiety, it's the uncertainty. It's stumbled a bunch of times. I think to be fair, some of the most interesting people, right, stumble quite a bit in life. Whether it's the external environment or the internal or whatever the universe brings. And in that I just know a lot of people just feeling and monitoring it right now because of everything AI is doing to our society, there's a lot of people just terrified and really anxious. And I think there's a lot of wisdom in what Ankur says. You could be paralyzed by that fear or you can imagine I already won. What would the person who has won do in this situation? And that alleviates the fear, it actually takes the gives you the the optimal path under uncertainty. Uh under heavy uncertainty under a lot of uncertainty. And don't use the old playbook, it be try to be the person that figures out how to write the new the next playbook.

我觉得 Max Tegmark 曾多次和我交谈，因为他是个狂野的人。他在职业生涯中做了很多奇怪而美丽的事情，但他也做了主流的事情。他说你必须两者都做一点。这样你仍然在系统内工作足够多，你是社区的一部分，你是流动的一部分，在那种情况下是科学和发展的流动。但也要做足够多的叛逆、狂野和奇怪的事情，来开辟自己的道路。两者都以某种方式，找到平衡，不要害怕有点奇怪。但也要变得非常熟练和善于成为社区的一部分，成为社区中受人尊敬的一部分。所以，两者都要。但这就像 Tom Waits 说的，我喜欢我的小镇带一点毒药。所以，一点点奇怪，或者有时很多奇怪，只是让他们成为一些美丽的人。

I think Max Max Tegmark has uh spoken with me times cuz he's a wild one. He's done a lot of weird and beautiful things in his career, but he also did the mainstream stuff. He said you have to do a little bit of both. So that you don't you still you still work enough within the system that you're part of the community, you're part of the flow of uh in that case it's science and the development of things. But also do enough renegade and wild and weird things to uh pave your own way. Both somehow and find that balance and not be afraid to be a little bit weird. But also become um uh very skillful and good at being part of a community and a respectful part of the community. So, both of those. But that that just like it's like uh Tom Waits said I like my town with a little drop of poison. So, just a little bit of weird or sometimes a lot of weird just makes them for some beautiful humans.

我认为社区这部分在未来实际上会变得更加重要。我们进行了大约 30 年的实验，看看我们能用电机器做多少事情？现在我们可以用机器做大量的事情，我希望我们会意识到我们实际上更喜欢和人一起做事。我们更喜欢飞过去面对面交谈。我们更喜欢进行口头讨论，而不是仅仅阅读论文。嗯，我认为我们的日常体验有一个峰值。所以，我的意思是，是的，我从我的科学工作和学术生涯中获得了很大的满足感，但回家和孩子们在一起，如果 FLT 问题解决了，去海滩和他们扔球，没有什么比这更好的了。就像那是我生命的意义。其他这些东西是锦上添花。但它的基础是家庭。是我关心的人和其他人在一起。

I think that community part is actually going to be even more important in the future. We've had this maybe 30-year experiment with how much can we do with machines? And now that we can do a ton with machines, my hope is that we're going to realize we actually prefer to do things with humans. We prefer to fly in and have an in-person conversation. We prefer to have an oral discussion instead of just reading a paper. Um I I think this there's like a peak to how much uh our day-to-day experience. So, I mean, in in terms of yes, I get a lot of fulfillment from my um scientific work and from my academic career, but going home to my kids and and going to the beach if the if FLT gets solved and just throwing a ball with them is there's nothing like that. Like that's the meaning of my life. This other stuff is the cherry on top. But like the foundation of it is is family. Is is people being around other people that I care about.

我认为随着 AI 越来越将智能商品化，我们会越来越意识到，智能和我们解决的所有不同难题只是锦上添花，或者旁边的樱桃。主要的是其他人

And I think as AI commodifies intelligence more and more, we'll realize that more and more that um in intelligence and all the all the different puzzles that we solve is is the just the cherry on top or a cherry to the side. The main thing is other humans

是的。

Yeah.

在我们的生活中。

in our life.

我无法告诉你，你们亲自来对话是多么荣幸。我已经成为你们三位的超级粉丝。感觉有点超现实，今天你们能和我交谈是很大的荣幸。这超级有趣。非常感谢。非常感谢，Hong Ge、Steve、Alex。这太棒了。

This is I can't tell you how much of an honor it is that you would uh come down in person for a conversation. I've become a huge fan of all three of you. It's feels a bit surreal and it's a big honor that you would um talk with me today. This was like super fun. Thank you so much. Thank you so much, Hong Ge, Steve, Alex. This was awesome.

感谢邀请我们。对我们来说也很有趣。感谢邀请我们。

Thanks for having us. A lot of fun for us, too. Thanks for having us.

这很棒。

This is great.