Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

发布时间 2025-06-14 20:15:34 来源

以下是亚历克斯·弗里德曼和陶哲轩（一位备受推崇的数学家，被誉为“数学界的莫扎特”）之间的一次对话。他们讨论了广泛的议题，包括高难度的数学问题、纳维-斯托克斯方程、计算机辅助证明、人工智能在数学中的作用以及素数的本质。陶哲轩首先讨论了Kakeya针问题，解释了它如何与偏微分方程、数论和波传播等多个领域相关联。他解释说，对它的研究与波的行为有关，特别是关于奇异点和纳维-斯托克斯方程（控制流体动力学）。他深入探讨了他发表的关于“平均三维纳维-斯托克斯方程”的论文，以及它如何试图理解流体中的奇异点是否能够形成。陶哲轩详细阐述了为什么难以证明纳维-斯托克斯方程的一般性质，他提到了麦克斯韦妖的概念，即即使在统计上不太可能，理论上也可能出现极其不可能的配置。他用这个概念来说明，即使存在会使流体趋于平静的粘滞力，流体的能量也可能集中到越来越小的尺度上。为了解决这个问题，陶哲轩讨论了他如何通过改变平均方程中的物理定律来设计一个爆破，一个能量达到奇点的强制情况。他提到这项工作提供了一种障碍，并排除了某些证明方式。他描述了解决纳维-斯托克斯问题的另一种方法，认为它可以通过构建一个液态图灵机来解决。他阐述说，水可以用来制造逻辑门，从而创建一个可以自我复制的机器。液态图灵机可能会导致爆破。陶哲轩指出，康威的生命游戏对这个想法产生了影响，系统的复制是这个构造的灵感来源。他还深入探讨了结构和随机性之间的二分法。陶哲轩随后将话题转向了数学、物理和工程之间的区别。物理学收集观察结果并提出模型，而数学则接受这些模型并描述由此产生的后果。数学从公理开始，并询问会得出什么结论，而科学则从一个问题开始。他说我们需要实验和理论两方面。讨论过渡到人工智能在数学中的作用，从计算机辅助证明和Lean形式证明编程语言开始。Lean能够逐行形式化代码，如果信任编译器，在数学上，就能 100% 保证论证的正确性。但人类的因素和模式识别可能会丢失。陶哲轩还提到了Alpha Proof，DeepMind 的 AI，它接受了对成功和失败证明进行强化学习的训练。进入数论领域，陶哲轩深入研究了素数的性质，素数通常被称为“数学的原子”。他解释了如何生成自然数，即加 1。素数，2 3 5 7 是乘法的。讨论还涵盖了孪生素数猜想的挑战。正如 Paul Radar 所说，虽然这是一个难题，但陶哲轩取得了一些进展，他解释说大约 60% 的统计输入会拖回来。后来，陶哲轩解释了庞加莱猜想，这是一个千禧年难题。他解释了这个挑战：如果任何三维物体都可以把它变成一个三维球体？如果可以降低维度，它就可以解决大量问题。对话还涉及了作为数学家的个人方面，包括情感投入的挑战以及在工作中找到正确平衡的重要性。陶哲轩最后分享了他对数学未来的希望。

This is a conversation between Alex Friedman and Terence Tao, a highly regarded mathematician known as the "Mozart of Math." They discuss a wide range of topics, including difficult math problems, Navier-Stokes equations, computer-assisted proofs, the role of AI in mathematics, and the nature of prime numbers. Tao begins by discussing the Kakeya needle problem, explaining how it connects to various areas like partial differential equations, number theory, and wave propagation. He explains how the study of it relates to the behavior of waves, specifically concerning singularities and the Navier-Stokes equations, which govern fluid dynamics. He dives into a paper he published concerning the "average three-dimensional Navier-Stokes equation," and how it tries to understand whether singularities in fluid can form. Tao elaborates on why it's difficult to prove general properties about the Navier-Stokes equations, invoking the concept of Maxwell's demon, which is the idea that extremely improbable configurations can theoretically emerge even though they are statistically unlikely. He uses this concept to illustrate how the energy of a fluid can potentially be concentrated into a smaller and smaller scale despite the presence of viscous forces, which would calm things down. To address this, Tao discusses how he engineered a blow-up, a forced situation where energy reaches a singularity, by altering the laws of physics in an average equation. He mentions that this work provides an obstruction and rules out certain ways to prove things. He describes another approach to the Navier-Stokes problem, suggesting that it might be resolvable by constructing a liquid Turing machine. He elaborates that water could be used to make logic gates, creating a machine that would replicate itself. The liquid touring machine could potentially lead to a blowup. Tao notes the influence of Conway's Game of Life on this idea, where the system replicates, as inspiration for this construct. He also dives into the dichotomy between structure and randomness. Tao then pivots to the difference between mathematics and physics, with engineering also in there. Physics gathers observations and suggests models, and mathematics takes the models and describes what consequences come from them. Mathematics starts with axioms and asks what conclusions come, while the sciences start with a question. He said one needs both experimental and theoretical. The discussion transitions to the role of AI in mathematics, starting with computer-assisted proofs and the Lean formal proof programming language. Lean enables formalizing a line by line code that mathematically, if you trust the compiler, gives a 100% guarantee that the arguments are correct. But the human element and pattern recognition can be lost. Tao also touches on Alpha Proof, the DeepMind AI that's trained on reinforcement learning on successful and failed proofs. Moving into the realm of number theory, Tao delves into the properties of prime numbers, often called the "atoms of mathematics." He explains how to generate the natural numbers, you can add 1. Primes, 2 3 5 7 are multiplicative. The challenges of the twin prime conjecture are also covered. While it's a struggle, as said by Paul Radar, Tao has made some progress by explaining that about 60% of the inputs in statistics will drag back. Later, Tao explains the Poincaré Conjecture, a Millennium Prize Problem. He explains the challenge: what if any three-dimensional object could you turn it into a three-dimensional sphere? If you can reduce dimensions, it solves a huge number of problems. The conversation also touches on the personal aspects of being a mathematician, including the challenges of emotional investment and the importance of finding the right balance in one's work. Tao concludes by sharing his hope for the future of mathematics.

摘要

Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the ...

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中英文字稿

The following is a conversation with Terrance Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of Math. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me, for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world.

以下是与特伦斯·陶的对话，他被广泛认为是历史上最伟大的数学家之一，常被称为“数学界的莫扎特”。他获得了菲尔兹奖和数学突破奖，并在数学和物理的众多领域做出了开创性的贡献。对我来说，这次对话是巨大的荣幸，原因有很多，包括特里在我们所有互动中表现出的谦逊和善良。这对我意义重大。

This is Alex Friedman podcast to support it. Please check out our sponsors in the description or at lexfriedman.com, search sponsors, and now dear friends, here's Terrance Tao. What was the first really difficult research level math problem that you encountered? One that gave you powers maybe. Well, I mean, in your undergraduate education, you learn about the really hard and possible problems like the Riemann hypothesis, the Trinparms conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable.

这是Alex Friedman的播客来支持它。请查看描述中的赞助商，或者访问 lexfriedman.com，搜索赞助商。现在，亲爱的朋友们，这里是Terrance Tao。你遇到的第一个真正艰难的研究级别的数学问题是什么？也许是让你获得能力的那个。在大学教育中，你会了解一些真正困难甚至近乎不可能的问题，比如黎曼假设和Trinparms猜想。你可以让问题变得任意复杂，这并不是真正的问题。实际上，我们甚至知道有些问题是无法解决的。

What's really interesting are the problems just to the boundary between what we can do better easily and what are hopeless. But what are problems where like existing techniques can do like 90% of the job and then you just need that remaining 10%. I think as a PhD student, the KKF problem certainly caught my eye and it just got solved actually. It's a problem I've worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Sujika Kea, like 1918 or so.

真正有趣的是那些介于我们能够轻松解决的问题与几乎无解的问题之间的边界问题。然而，有一些问题是现有技术可以完成90%的工作，然后只需解决剩下的10%。我认为，作为一名博士生，KKF问题确实引起了我的注意，而且实际上这个问题最近刚被解决。这是我早期研究时投入了大量精力的问题。从历史上看，它源自于1918年左右日本数学家Sujika Kea提出的一个小难题。

So the puzzle is that you have a needle on the plane. I think like driving other icon on a road. You want it to execute a U-turn. You want to turn the needle around. But you want to do it as little space as possible. You want to use this little area in order to turn it around. But the needle is infinitely maneuverable. You can imagine just spinning it around. It's as a unit needle. You can spin it around its center. That gives you a disk of area I think PIVO 4.

这个问题是这样的：你在平面上有一根针。我想就像在路上驾驶其他图标一样。你想让它执行一个掉头动作，把针掉转过来。但你希望尽可能少占用空间来完成这个动作。你想利用这个小区域来完成掉头。然而，这根针是可以无限制操控的，你可以想象把它绕着中心旋转。这根针被视为单位针，你可以绕着它的中心旋转。这会形成一个面积为Pi/4的圆盘。

Or you can do a 3.U-turn, which is why we teach people in the driving schools to do. And that actually takes area PIVO 8. So it's a little bit more efficient than a rotation. And so for a while people thought that was the most efficient way to turn things around. But as a COVID showed that in fact you could actually turn the needle around using as little areas you wanted. So 0.001, there was some really fancy multi back and forth U-turn thing that you could do.

或者你可以执行一个三点掉头，这也是我们在驾驶学校教大家的技巧。这样做实际上需要PIVO 8这个区域。因此，相较于旋转，这种方法更加高效。一段时间以来，人们认为这是最有效的掉头方式。但是，COVID显示你实际上可以在更小的区域内掉头，比如0.001，就可以通过一些非常巧妙的来回U形转弯完成。

That you could turn the needle around. And so doing it would pass through every intermediate direction. Is this in the traditional plane? This is an international plane. So we understand everything in two dimensions. So the next question is what happens in three dimensions. So suppose like the Hubble Space Telescope is tube in space. And you want to observe every single star in the universe. So you want to rotate the telescope to every single direction.

你可以旋转这个针，它会经过所有中间的方向。这是在传统的平面上吗？这是一个国际平面。因此，我们在二维中理解一切。所以下一个问题是，三维中会发生什么。比如说，哈勃太空望远镜是太空中的一个管道，你想观察宇宙中的每一颗星星，因此你希望将望远镜旋转到每个方向。

And he's unrealistic part. Suppose that space is at a premium, which totally is not. You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky. How small a volume do you need to do that? And so you can modify the physical bagage's construction. And so if your telescope has zero thickness, then you can use as little volume as you need. That's a simple modification of the two-dimensional construction.

他有些不现实的部分。假设空间非常紧张，但实际上并不是。你希望占用尽可能少的空间来旋转你的望远镜，以便观察到天空中的每一颗星星。那么需要多小的空间才能做到这一点呢？因此，你可以对物理设备的结构进行修改。如果你的望远镜是没有厚度的，那么你可以使用所需的最小空间。这是对二维结构的简单改动。

But the question is that if your telescope is not zero thickness but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta. So as delta gets smaller as you need to get thinner, the volume should go down. But how fast does it go down? And the conjecture was that it goes down very, very slowly, like log with it going, roughly speaking. And that was proved after a lot of work.

问题在于，如果你的望远镜不是完全没有厚度，而是非常非常薄，具有一些厚度δ，那么要能够看到每个方向所需的最小体积是多少，作为δ的一个函数。当δ变得更小、更薄时，体积应该减小。但它减小的速度有多快？有人猜测这个缩减速度非常非常慢，大致来说，就像对数一样。经过大量的研究工作，这个猜测得到了证明。

So this seems like a puzzle-wise interesting. So it turns out to be surprisingly connected to a lot of problems in posh differential equations, in number theory, in geometry, commentarics. For example, in wave propagation, you splash some water around, you create water waves, and they travel in various directions. But waves exhibit both particle and wave type behavior. So you can have what's got a wave packet, which is like a very localized wave that is low-class in space and moving a certain direction in time.

这看起来像是一个从谜题的角度来看很有趣的问题。事实上，它与偏微分方程、数论、几何学以及组合数学中的许多问题有着意外的联系。例如，在波的传播中，当你把水溅起时，会产生水波，它们向不同的方向传播。但波既表现出粒子性质，又表现出波的性质。比如，你可以有一个所谓的波包，它是一种非常局部化的波，在空间中范围很小，并且随着时间向某个特定方向移动。

And so if you plot it into space and time, it occupies a region, which looks like a tube. And so what can happen is that you can have a wave, which initially is very dispersed, but it all focuses at a single point later in time. Like you can imagine dropping a pebble into a pond and the ripple spread out. But then if you time reverse that scenario and the equations are way more than a time reversible, you can imagine ripples that are converging to a single point, and then a big splash occurs, maybe even a singularity. And so it's possible to do that.

如果把它放入时空中，你会发现它占据了一个类似于管状的区域。这个过程中可能发生的情况是，一个起初非常分散的波在之后的某个时间点会集中在一起。就像你可以想象将一颗小石子扔进池塘，涟漪四处扩散。但如果你把这个场景时间倒流，并且方程完全可以实现时间可逆，你就能想象这些涟漪聚集到一个点，然后出现巨大的飞溅，甚至可能形成一个奇点。所以，这是有可能实现的。

And geometric was going on is that there's always light rays. So like if this wave represents light, for example, you can imagine this wave as a superposition of photons, all traveling at the speed of light. They all travel on these light rays, and they're all focusing at this one point. So you can have a dispersed wave focus into a very concentrated wave at one point in space and time, but then it's deflacuses again, and it separates.

几何上，这意味着总有光线在传播。如果这个波代表光，比如说，你可以想象这个波是由许多光子叠加而成的，这些光子都以光速传播。它们都沿着这些光线传播，并且在一个点上聚焦。所以，一个分散的波可以在空间和时间的某一点聚集成一个非常集中的波，但随后它会再次发散并分离。

But potentially if the conjunctia had a negative solution, so what I mean is that there's a very efficient way to pack tubes pointing to different directions into a very, very narrow region of very narrow volume. Then you would also be able to create waves that start out, there'll be some arrangement of waves that start out very, very dispersed, but they would concentrate not just at a single point, but there'll be a large, there'll be a lot of concentrations in space and time.

如果“结合理论”存在一个负面解决方案，我的意思是说，有一种非常高效的方法可以将指向不同方向的管子打包到一个非常狭窄的空间里。那么，你也可以创造出一种波动，这些波动一开始会非常分散，但它们不仅会集中在一个点上，而是在空间和时间上有大量集中。

And you could create what's called a blow-up where these waves, they have to do become so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear. And so in mathematical physics, we care a lot about whether certain equations and wave equations are stable or not, whether they can create these singularities. There's a famous, I saw a problem called the Navier-Stokes regularity problem. So the Navier-Stokes equations, equations that govern a fluid flow or incompressible fluid is like water.

你可以制造出所谓的“爆发”现象，这种情况下波浪会变得如此强烈，以至于支配它们的物理定律不再是简单的波动方程，而是更复杂且非线性的体系。所以在数学物理学中，我们非常关心某些方程和波动方程是否稳定，是否会产生这些奇点。有一个著名的问题叫做纳维-斯托克斯方程的正则性问题。纳维-斯托克斯方程支配着流体运动，主要描述像水这样不可压缩的流体的流动。

The question asks, if you start with a smooth velocity fluid of water, can it ever concentrate so much that the velocity can be infinite at some point? That's got a singularity. We don't see that in real life, if you splash around water and the bathtub we want to explode on you, or have water leaving at a speed of light, but potentially it is possible. And in recent years, the consensus has drifted towards the belief that in fact for certain very special initial configurations of say water, that singularities can form.

这个问题是在询问，如果你从一股流动平稳的水开始，它是否可能会逐渐集中到某一点，使得速度在那一点上变得无限大？这就是一个奇点。在现实生活中，我们并没有见到这种现象，比如当你在浴缸里拍打水时，水不会突然爆炸或以光速飞溅出去。不过，从理论上讲，这是可能的。近年来，研究的共识逐渐倾向于相信，对于某些非常特殊的初始条件下，比如一定配置的水，奇点确实有可能形成。

But people have not yet been able to actually establish this. The Clay Foundation has these seven millennium prize problems, has a million dollar prize for solving one of these problems. So this is one of them. Of these seven, only one of them has been solved at the point where you can check your experiment. So the Kekena conjecture is not directly related to the Navier-Stokes problem, but understanding it would help us understand some aspects of things like wave concentration, which would indirectly probably help us understand the Navier-Stokes problem better.

然而，人们尚未能够真正解决这个问题。克雷数学研究所设立了七个千禧年大奖难题，解决其中一个问题就可以获得一百万美元的奖赏。这是其中的一个。到目前为止，这七个难题中只有一个已经被解决，可以进行实验验证。Kekena 猜想与纳维-斯托克斯问题没有直接关系，但理解Kekena 猜想将有助于我们更好地理解一些现象，比如波的集中，这可能会间接帮助我们更好地理解纳维-斯托克斯问题。

Can you speak to the Navier-Stokes? So the existence of smoothness, like you said, millennial prize problem. You made a lot of progress on this one. In 2016, you published a paper finite time blow-up for an average three-dimensional Navier-Stokes equation. So we're trying to figure out if this thing usually doesn't blow up. But can we say for sure it never blows up? Right. That is literally the moving dollar question.

您能谈谈纳维-斯托克斯方程吗？就像您提到的，关于其光滑性存在问题，这是个千禧年大奖问题。您在这一领域取得了很大进展。2016年，您发表了一篇关于三维纳维-斯托克斯方程在有限时间内爆发的论文。我们正在试图弄清楚这个方程是否通常不会爆发。但我们能否确定它永远不会爆发呢？是的，这确实是一个价值连城的问题。

This is what distinguishes mathematicians from pretty much everybody else. If something holds an act 9.99% of the time, that's good enough for most things. But mathematicians, on a few people who really care about whether, like, 100% really 100% of all situations are covered by, most of the time, water does not blow up, but could you design a very special initial state that does this? And maybe we should say that this is a set of equations that govern in the field of fluid dynamics.

这就是数学家与其他人的区别所在。如果某件事情在99.99%的情况下成立，对于大多数事情来说，这已经足够好了。但是数学家却是少数真正关心是否100%在所有情况下都成立的人。对于他们来说，虽然大多数情况下水不会爆炸，但是否可以设计一个非常特殊的初始状态让它这么做？也许我们应该说，这是一组用于流体力学领域的方程。

Yeah, trying to understand how fluid behaves and it's actually trying to be really complicated. You know, fluid is extremely complicated thing to try and amount. Yeah, so it has practical importance. So this clay prize problem concerns what's called the incompressible Navier-Stokes, which governs things like water. There's something called the compressible Navier-Stokes, which governs things like air.

是的，试图理解流体的行为实际上是非常复杂的。你知道，流体是一个极其复杂的东西，很难真正弄清楚。它确实有实际的重要性。这个克莱数学奖的问题涉及所谓的不可压缩的Navier-Stokes方程，该方程用于描述像水这样的流体行为。而还有一种叫做可压缩的Navier-Stokes方程，用于描述像空气这样的流体行为。

And that's particularly important for weather prediction. Weather prediction, it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier-Stokes equations as best they can. Also gathering a lot of data to let they can get, they can initialize the equation. There's a lot of moving parts. So it's very important from practically.

这对天气预报来说尤为重要。天气预报中，计算流体动力学占了很大一部分。很大程度上，它实际上是在尽可能解决纳维-斯托克斯方程。同时，需要收集大量数据以初始化这些方程。整个过程包含很多环节，因此从实际角度来看，这非常重要。

Why is it difficult to prove general things about the set of equations like in that blowing up? Short answer is Maxwell's demon. So Maxwell's demon is a concept in thermodynamics. Like if you have a box of two gases in an oxygen and nitrogen, and maybe you start with all the oxygen on one side and nitrogen the other side, but there's no barrier between them. Then they will mix. And they should stay mixed. There's no reason why they should unmix.

为什么很难证明关于这样一组方程的一般性问题，比如爆炸呢？简单回答就是麦克斯韦妖。麦克斯韦妖是一个热力学中的概念。想象一下，你有一个盒子，里面有氧气和氮气，可能刚开始氧气在一边，氮气在另一边，但中间没有隔板。那么它们会混合在一起，并且应该保持混合状态，没有理由使它们重新分开。

But in principle, because of all the collisions between them, there could be some sort of weird conspiracy. Like maybe there's a microscopic demon, called Maxwell's demon, that will every time an oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen drifts on to one side and the nitrogen goes to the other.

原则上，由于它们之间的所有碰撞，可能会发生某种异常现象。比如，可能存在一个叫做麦克斯韦妖的微观小妖怪，会在氧原子和氮原子每次碰撞时，让它们以某种方式弹开，从而使氧原子聚集到一边，而氮原子聚集到另一边。

And you could have an extremely improbable configuration emerge, which we never see. And we're statistically extremely unlikely. But mathematically, it's possible that this can happen. And we can't wall it out. And this is a situation that shows up a lot in mathematics. A basic example is the digits of pi. 3, 4, 1, 4, 1, 5, and so forth.

这段话大致的意思是：有时可能会出现一些非常不可能的情况，尽管我们几乎从未见过这种情况，从统计学上看，它发生的概率极低，但从数学的角度来看，这种情况是可能的，我们不能完全排除它。这样的情况在数学中经常出现。一个基础的例子就是圆周率（π）的数字，例如3、1、4、1、5等等。

The digits look like they have no pattern. And we believe they have no pattern. On the long term, we should see as many ones and 2s and 3s as 4s and 5s and 6s. There should be no preference in the digits of pi to favor, let's say, 7 over 8. But maybe there's some demon in the digits of pi that every time you can beat more more digits, it's a biases one digit to another.

这些数字看起来没有任何规律。我们相信它们确实没有规律。从长远来看，应该看到1、2、3和4、5、6出现的频率是相同的。也就是说，圆周率的数字中不应该有某个数字比其他数字更常出现，比如7不应该比8更频繁。但或许在圆周率的数字中存在某种“恶魔”，每当你计算更多的数字时，它就会使某个数字出现的频率偏向于另一个。

And this is a conspiracy that should not happen. There's no reason it should happen. But there's no way to prove it with our current technology. Okay, so getting back to Navier Stokes, a fluid has a certain amount of energy. And because if fluid is in motion, the energy gets transported around. And what is also viscous?

这是一个不应该发生的阴谋。没有理由让它发生。但以我们目前的技术，这种情况无法证明。好的，回到Navier-Stokes问题，流体具有一定的能量。由于流体在运动中，能量被传递。而且还有粘性的问题。

So if the energy is spread out over many different locations, the natural viscosity of fluid will just damp out the energy and it will go to zero. And this is what happens when we actually experiment with water. I get it, you splash around, there's some turbulence and waves and so forth. But eventually it settles down and the lower the amplitude, the smaller velocity the more calm it gets.

因此，如果能量分散在许多不同的位置，液体的自然黏性会将能量逐渐消耗殆尽，最后能量会趋于零。这就是我们在实际进行水实验时观察到的现象。我明白，当你搅动水的时候，会出现一些湍流和波浪等现象。但最终水面会平静下来，波浪的幅度越小，速度越慢，水面就会越平静。

But potentially there is some sort of demon that keeps pushing the energy of the fluid into a smaller and smaller scale. And we move faster and faster and faster speeds the effect of viscosity is relatively less. And so it could happen that it creates some sort of a sort of similar blob scenario where the energy of fluid starts off at some large scale and then it all sort of transfers energy into a smaller region of the fluid, which then at a much faster rate moves into an even smaller region and so forth.

可能有某种“恶魔”在不断地将流体的能量推向越来越小的尺度。当我们移动速度越来越快时，粘性的效果相对较小。因此，这可能导致一种类似于“能量团”的情况，其中流体的能量在一开始分布在较大的尺度上，然后这些能量逐渐转移到流体的较小区域，随后以更快的速度进入更加微小的区域，依此类推。

And each time it does this, it takes maybe half as long as the previous one. And then you could actually converge to all the energy concentrating in one point in a finite amount of time. And that's an hour's go to finite hand blow up. So in practice this doesn't happen. So water is what's called turbulent. So it is true that if you have a big eddy of water, it will tend to break up into smaller eddies.

每次它这样做时，所需的时间可能只是上一次的一半。最终，所有能量会在有限的时间内集中到一个点上。然而，这在现实中并不会真正发生。水是被称为湍流的状态。也就是说，当水中出现一个大型漩涡时，它往往会分裂成更小的漩涡。

But it won't transfer all this energy from one big eddy into one smaller eddy, it will transfer into maybe three or four. And then those ones split up into maybe three or four small eddies of their own. And so the energy dispersed to the point where the viscosity can then keep a thing under control. But if it can somehow concentrate all the energy, keep it all together and do it fast enough that the viscous effects don't have enough time to come everything down, then this will all kind of go.

但这种能量并不会从一个大漩涡直接转移到一个小漩涡中，而是可能分散到三个或四个小漩涡中。然后，这些小漩涡又各自分裂成另外三个或四个更小的漩涡。因此，能量逐渐分散至粘性作用能够控制的范围内。然而，如果能以某种方式将所有能量集中在一起，并且速度足够快，以至于粘性作用无法及时影响整个过程，那么这一切就会迅速发生。

So there were papers who had claimed that, oh, you just need to take into account conservation of energy and just carefully use the viscosity and you can keep everything under control for not just an aviastokes, but for many, many types of equations like this. And so in the past there have been many attempts to try to obtain what's called global regularity for naviastokes, which is the opposite of finite hand blow up that have lost you say smooth.

有些论文声称，只要考虑能量守恒，并仔细使用粘性，就可以不仅仅在处理Navier-Stokes方程时，还可以对许多类似的方程进行有效控制。因此，过去曾有许多尝试试图获得所谓的Navier-Stokes方程的全局正则性，这与有限时间内发生爆破解是不一样的，这意味着解决方案保持平滑。

And it all failed. There was always some sin error, some subtle mistake, and it couldn't be salvaged. So what I was interested in doing was trying to explain why we were not able to disprove finite hand blow up. I couldn't do it for the actual equations of fluids, which were too complicated. But if I could average the equations of motion of naviastokes, basically if I could turn off certain types of ways in which water interacts, and only keep the ones that I want.

所有的努力都失败了。总是有一些错误存在，一些细微的错误，结果无法挽回。所以，我感兴趣的是尝试解释为什么我们无法证伪有限手爆炸。我无法处理真正的流体方程，因为它们过于复杂。但如果我能对Navier-Stokes方程的动力进行平均化，基本上，如果我能关闭某些类型的水相互作用方式，只保留我想要的那些。

So in particular, if there's a fluid and it could transfer this energy from a large 80 into this small 80 or this other small 80, I would turn off the energy channel that would transfer energy to this one and direct it only into this smaller 80. While still preserving the law of conservation energy. So you try and make a blow up. Yeah. Yeah. So I basically engineer a blow up by changing the laws of physics, which is one thing that mathematicians are allowed to do. We can change the equation.

所以，特别是，如果有一种流体可以将能量从一个大容器80转移到一个小容器80或另一个小容器80，我会关闭通往这个容器的能量通道，只把能量引导进这个小容器80，并且仍然遵守能量守恒定律。所以，你试图制造一个爆炸。对，没错。基本上，我通过改变物理法则来设计一个爆炸，而这正是数学家可以做的事情。我们可以改变方程。

How does that help you get closer to the proof of something? Right. So it provides what's called an obstruction in mathematics. So what I did was that basically if I turned off the certain parts of the equation, which usually when you turn off certain interactions make it less nonlinear, it makes it more regular and less likely to blow up. But I found that by turning off a very well-designed set of interactions, I could force all the energy to blow up in finite time.

这怎样帮助你更接近某件事情的证明呢？好的，这在数学中被称为“障碍”。我所做的大致是，如果我关闭方程中的某些部分，通常关闭某些交互会使其更趋向线性，并且更规律，较不容易失控。然而，我发现通过关闭一组设计得非常巧妙的交互，我可以迫使所有的能量在有限的时间内失控。

So what that means is that if you wanted to prove global regularity for Navier-Stokes for the actual equation, you must use some feature of the true equation which my artificial equation does not satisfy. So it rules out certain approaches. So the thing about math is it's not just about finding a technique that is going to work in applying it, but you need to not take the techniques that don't work.

这句话的意思是，如果你想为纳维-斯托克斯方程的全局正则性做证明，你必须利用实际方程的一些特征，而这些特征是我的人工方程所不具备的。因此，这就排除了一些方法。所以数学的问题不仅仅在于找到一个有效的技术并应用它，还在于避免使用那些无效的技术。

And for the problems that are really hard, often there are dozens of ways that you might think might apply to solve the problem. But it's only after a lot of experience that you realize there's no way that these methods are going to work. So having these counterexamples for nearby problems kind of rules out. It saves you a lot of time because you're not wasting energy on things that you now know cannot possibly ever work.

对于那些真正困难的问题，通常会有很多方法让你觉得可以用来解决问题。但只有经过大量经验积累后，你才会意识到这些方法根本行不通。因此，拥有这些类似问题的反例就显得很重要。它帮你排除了一些路子，节省了大量时间，因为你不再在那些你已经知道绝对行不通的方法上浪费精力。

How deeply connected is it to that specific problem of fluid dynamics or is it some more general intuition you build up about mathematics? Right. Yeah. So the key phenomenon that my technique exploits is what's called supercriticality. So in positive and differential equations, often these equations are like a tug of war between different forces. So in Navier-Stokes, there's the dissipation force coming from viscosity and it's very honest to it.

这与流体动力学的具体问题有多大关系？或者是一种你对数学建立的更普遍的直觉？对。是的。我使用的技术关键在于一个称为超临界性的现象。在偏微分方程中，这些方程常常像是在不同力之间拔河。例如，在Navier-Stokes方程中，有来自粘性的耗散力，而它对此有着非常直接的影响。

It's linear. It calms things down. If viscosity was all there was, then nothing bad would ever happen. But there's also transport that energy from one location of space can get transported because of fluid in motion to other locations. And that's a nonlinear effect and that causes all the problems. So there are these two competing terms in the Navier-Stokes equation, the dissipation term and the transport term.

这是线性的。它能使事情平静下来。如果只有粘性，那么就不会发生任何坏事。但由于流体的运动，能量可以从一个地方传输到另一个地方，这就是非线性效应，会导致各种问题。因此，在纳维-斯托克斯方程中存在这两个相互竞争的项：耗散项和传输项。

If the dissipation term dominates, if it's large, then basically you get regularity. And if the transport term dominates, then we don't know what's going on. It's a very nonlinear situation. It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales or vice versa. So Navier-Stokes is also supercritical. So at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.

如果耗散项占主导地位，也就是说它很大，那基本上系统会表现出规则性。而如果传输项占主导地位，我们就不太清楚会发生什么。这种情况下表现得非常非线性、不可预测和湍流。所以有时候这些力在小尺度上是平衡的，但在大尺度上或反之却不平衡。纳维-斯托克斯方程也是超临界的，在越来越小的尺度下，传输项比黏性项要强得多。

So the viscosity terms are things that calm things down. And so this is why the problem is hard. In two dimensions, the Soviet methodical and the addition of the sky, she in the 60s shows that in two dimensions, there was no blow up. And in two dimensions, the Navier-Stokes equations is what's called critical, the effect of transport and the effect of viscosity, apart the same strength, even at very, very small scales.

粘性项是用来平息事物的因素。这也是问题难以解决的原因。在二维情况下，苏联在20世纪60年代的系统研究和突破表明，在二维情况下，不会出现不稳定或“爆炸”的现象。对于二维的纳维-斯托克斯方程，情况被称为临界状态，运输效应和粘性效应的力量相当，即使在非常非常小的尺度下也是如此。

And we have a lot of technology to handle critical and also subcritical equations and proof of regularity. But for supercritical equations, it was not clear what was going on. And I did a lot of work and then there's been a lot of follow-up showing that for many other types of supercritical equations, you can create all kinds of blow-up examples. Once the nonlinear effects dominate the linear effects at small scales, you can have all kinds of bad things happen.

我们拥有许多技术可以处理关键性和亚关键性的方程，以及证明其规律性。但是，对于超关键性的方程，还不清楚会发生什么。我进行了大量研究，随后有很多后续研究表明，对于许多其他类型的超关键性方程，可以创造出各种爆炸性例子。当非线性效应在小尺度上占主导地位时，各种不良情况都有可能发生。

So this is sort of one of the main insights of this line of work is that supercriticality versus criticality and subcriticality. This makes a big difference. I mean, that's a key qualitative feature that distinguishes some equations for being nice and predictable and like planetary motion. I mean, there are certain equations that you can predict for millions of years or thousands at least.

这一研究领域的主要见解之一是超临界、临界和次临界的区别。这种区别非常重要。我指的是，这是一个关键的定性特征，它将某些方程从容易和可预测的，比如行星运动的方程中区分开来。我的意思是，有些方程可以用于预测数百万年或至少数千年的周期。

Not really a problem, but there's a reason why we can't predict the weather past two weeks into the future because there's a supercritical equation. Lots of really strange things are going on at very fine scales. So whenever there is some huge source of nonlinearity that can create a huge problem for predicting what's going to happen.

这实际上不是个问题，但有个原因让我们无法预测两周以后的天气，因为存在一个超临界方程。在非常微小的尺度上发生了许多非常奇怪的现象。因此，每当有一个巨大的非线性因素出现时，它就会给预测未来可能发生的事情带来极大的困难。

Yeah. And if nonlinearity is somehow more and more featured and interesting at small scales, I mean, there are many equations that are nonlinear, but in many equations, you can approximate things by the bulk. So for example, planetary motion, if... you want to understand the orbit of the moon or Mars or something, you don't really need the microstructure of like the seismology of the moon or exactly how the mass is distributed. You can also approximate these patterns by point masses and just the aggregate behavior is important. But if you want to model a fluid like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this for supercritical equations, the fine-scaled information is really important.

好的。如果在小尺度上，非线性表现得越来越突出和有趣，其实有很多方程是非线性的，但在许多方程中，你可以通过总体来进行近似。比如说，行星运动，如果你想了解月球或火星的轨道，你并不需要知道月球的地震学微观结构或物质分布的具体情况。你可以将这些模式近似为点质量，只要了解总体行为就够了。但如果你要模拟流体，比如天气，你不能仅仅说洛杉矶的温度是多少，风速是多少。对于超临界方程，细致的信息就变得非常重要。

If we can just linger on the Navier-Stokes equations a little bit. So you've suggested maybe you can describe it that one of the ways to solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer. Right. And then show that the halting problem from competition theory has consequences for fluid dynamics. So show it in that way. Can you describe this?

如果我们可以稍微多花一点时间来讨论纳维-斯托克斯方程。你提到过，也许可以通过某种方式来解决它，或者用一种消极的方式解决它，那就是构建一种液体计算机。对，然后展示竞争理论中的停机问题如何对流体动力学产生影响。可以这样来说明这一点吗？

Yeah. So this came out of all this work of constructing this average equation that blew up. So as part of how I had to do this, so this is the naive way to do it. You just keep pushing every time you get an energy at one scale, you push it immediately to the next scale as fast as possible. This is the naive way to force blow up. In terms of in 5 and high dimensions this works. But in three dimensions, there was this funny phenomenon that I discovered that if you keep, if you change loads of physics, you just always keep trying to push the energy into small, small scales.

好的。这段话是从构建一个失控的平均方程的过程中产生的。对于如何做到这一点，我采用的是一个简单的方法。每当一个尺度上出现能量时，你就尽可能快地将其转移到下一个尺度。这是迫使系统失控的简单方法。在五维和更高维度中这是有效的，但在三维中，我发现了一个有趣的现象。如果你不断改变许多物理参数，并总是试图把能量压入更小的尺度，就会出现这种现象。

What happens is that the energy starts getting spread out into many scales at once. So you have energy at one scale, you're pushing it into the next scale, and then as soon as it enters that scale, you also push it to the next scale, but there's still some energy left over from the previous scale. You're trying to do everything at once. And this spreads out the energy too much. And then it turns out that it makes it vulnerable for viscosity to come in and actually just damp out everything. So it turns out this directive, which doesn't actually work.

发生的情况是，能量同时被分散到多个层级上。也就是说，当你在一个层级上有能量时，你会把它推动到下一个层级，然后一旦进入新的层级，你又将其推向再下一个层级，但之前的层级还会有一些能量剩余。这样你试图同时做很多事情，能量就被分散得太多了。结果，这种分散使得能量容易被粘性耗散掉。最终，事实证明这种做法其实是行不通的。

It was a separate paper by some of the authors that I actually showed this in three dimensions. So what I needed was to program at the lay, so kind of like air locks. So I needed an equation which would start over a fluid doing something at one scale. It would push this energy into the next scale, but it would stay there until all the energy from the larger scale got transferred. And only after you pushed all the energy in, then you sort of open the next gate, and then you you push that in as well.

这其实是某些作者发表的另一个独立论文中展示的三维概念。我需要的是编写一个类似于气闸的程序，也就是说，我需要一个方程来在一个尺度上启动某种流体的活动。这个方程会将能量推动到下一个尺度，但是需要等到所有来自较大尺度的能量都被转移完毕后，才能打开下一个闸门，继续推动那个尺度的能量流动。

So by doing that, the energy enters forward, scaled by scale, in such a way that it's always localized at one scale at a time. And then it can resist the effect of viscosity because it's not dispersed. So in order to make that happen, I had to construct a rather complicated non-linearity. And it was basically like, it was constructing like an electronic circuit. So I actually thank my wife for this because she was trained as an electro engineer.

通过这样的操作，能量按比例逐级进入，使其每次都局限在一个特定的尺度上。这样做可以抵抗粘度的影响，因为能量不会被分散。为了实现这个目标，我必须设计一个相当复杂的非线性结构。基本上，这就像在构建一个电子电路。所以我非常感谢我的妻子，因为她是电子工程师培训出身。

And you know, she talked about, you know, she had to design circuits and so forth. And you know, if you want a circuit that does a certain thing, like maybe have a light, that flashes on and then turns off and then on and off, you can build it from more primitive components, you know, capacitors and resistors and so forth. And you have to build a diagram. You can sort of follow up your eyeballs and say, oh yeah, the current will build up here and it will stop and then it will do that.

你知道的，她谈到她需要设计电路等等。如果你想要一个电路实现某个功能，比如让一盏灯闪一下，然后关掉，再反复打开和关闭，你可以用一些更基础的组件来实现，比如电容器和电阻器等等。你需要画一个电路图，你可以通过观察这个图说，“哦，电流会在这里积累，然后停止，再继续运行。”

So I knew how to build the analog of basic electronic components. You're like resistors and capacitors and so forth. And I would stack them together in such a way that I would create something that would open one gate and then there would be a clock, and then once the clock hits us in first, I would have closed it. It kind of a root goberg type machine, but described mathematically. And this ended up working.

所以我知道如何构建基本电子元件的模拟，比如电阻和电容等等。我会把它们以某种方式组合在一起，这样就能创造出一个装置：首先打开一个开关，然后通过时钟控制，一旦时钟达到我们预设的时间，我就会关闭它。这有点像鲁布·戈德堡式的机器，但用数学方式来描述。最终，这个方法真的奏效了。

So what I realized is that if you could pull the same thing off for the actual equations, so if the equations of water support a computation, so like if you can imagine kind of a steam punk, but it's really water punk type of thing where, you know, so modern computers are electronic, you know, they're powered by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of water moving in a certain velocity, and maybe it's there are two different configurations corresponding to a bit being up or down.

所以我意识到，如果你能在真实的方程式上实现类似的操作，那么如果水的方程式支持一种计算，就像一种“水朋克”的概念。现代计算机是电子产品，由电子通过非常微小的电线并与其他电子相互作用来驱动。而在这种“水朋克”中，你可以想象成水的脉冲以某种速度移动，可能有两种不同的配置对应于一个比特的上或下。

Probably if you had two of these moving bodies of water collide, they would come out with some new configuration, which would be something like an AND gate or OR gate. You know, the output would depend on a very particular way on the inputs. And like you could chain these together and maybe create a two-in-machine and then you have computers, which I made completely out of water. And if you have computers, then maybe you could do robotics, so hydraulics and so forth. And so you could create some machine, which is basically a fluid analog, what's called a VONOMIN machine.

可能如果你让两个移动的水体碰撞，它们会形成某种新的结构，类似于一个与门（AND gate）或或门（OR gate）。这意味着输出会非常依赖于输入的特定方式。而且你可以将这些结构连接在一起，可能就能创造一个两输入的机器，从而用水完全构建一个计算机。如果你有了计算机，那么或许就可以进行机器人控制，因此能应用到液压系统等领域。这样，你可能就可以创造出一种机器，实际上是一个流体版的冯·诺依曼（VONOMIN）机器。

So a VONOMIN proposed, if you want to colonize Mars, the sheer cost of transporting people on machines to Mars is just ridiculous. But if you could transport one machine to Mars, and this machine had the ability to mine the planet, create some more materials, to smell them, and build more copies of the same machine, then you could colonize the whole planet over time. So if you could build a fluid machine, which is a fluid robot, and what it would do, its purpose in life, it's programmed so that it would create a smaller version of itself, in some sort of cold state, it wouldn't start just yet.

因此，一位名叫VONOMIN的人提出了一个建议：如果你想殖民火星，把人类使用机器运送到火星的成本真是太高了。但是，如果你能够运送一台机器到火星，而这台机器具备挖掘火星资源、制造更多材料、冶炼资源以及复制更多同类机器的能力，那么随着时间的推移，你就可以完成整个星球的殖民。因此，如果你能够制造一种流体机器，也就是流体机器人，这种机器的使命就是被编程成能够创建一个更小版本的自己，并且在一种冷状态下等待启动，那就能实现这样的目标。

Once it's ready, the big robot configured water would transport all the energy into the smaller configuration and then power down. And then I'd clean myself up. And then what's left is this newest set, which would then turn on, and do the same thing, but smaller and faster. And then the equation has a certain skating symmetry. Once you do that, it can just keep iterating. So this, in principle, would create a blur for the actual Navier Stokes. And this is what I managed to accomplish for this average Navier Stokes. So it provided this sort of roadmap to solve the problem.

一旦准备就绪，大型机器人配置的水将把所有能量传输到较小的配置中，然后关闭。而后，我会清理自己。剩下的是最新的一组，它会启动，并以更小、更快的方式做同样的事情。接着，方程式具有某种滑动对称性。一旦完成这一操作，它就可以不断迭代。因此，从原理上来说，这会给实际的Navier-Stokes创造一种模糊效果。而我成功地在平均Navier-Stokes方程中实现了这一点。这为解决该问题提供了某种路线图。

Now this is a pipe dream, because there are so many things that are missing for this to actually be a reality. So I can't create these basic logic gates. I don't have these special conductors of water. I mean, there's candidates that include vortex rings that might possibly work, but also, you know, analog computing is really nasty, like a bit of digital computing. I mean, because there's always errors, you have to do a lot of error correction along the way. I don't know how to completely power down the big machine, so it doesn't interfere with the writing of the smaller machine.

这是一个遥不可及的梦想，因为要让它成为现实，仍有太多东西缺失。我无法创造这些基本的逻辑门电路，也没有这些特殊的水导体。我知道有些候选方案，比如可能有效的涡流环，但模拟计算真的很麻烦，就像数字计算一样。因为总是有误差，你必须不断地进行误差修正。我也不知道如何彻底关闭大型机器，以免它干扰小型机器的运作。

But everything in principle can happen. Like it doesn't contradict any of the laws of physics. So it's sort of evidence that this thing is possible. There are other groups who are now pursuing ways to make Navier Stokes blow up, which are nowhere near as ridiculously complicated as this. They actually are pursuing much closer to the direct self-similar model, which can, it doesn't quite work as it is, but there could be some simpler scheme than what I just described to make this work. There is a real leap of genius here to go from Navier Stokes to this toying machine.

原则上，一切皆有可能。就像这并不与任何物理定律相矛盾一样。所以这就像是某种证据，证明这件事是可能的。现在有其他团队正在研究如何让纳维-斯托克斯方程"爆炸"的方法，这些方法远没有我刚才描述的那样复杂。他们实际上正在探索更接近直接自相似模型的方法，虽然目前还不完全可行，但可能有比我刚才描述的更简单的方法来实现它。在从纳维-斯托克斯方程到这个尝试的模型之间，确实存在一个天才的飞跃。

So it goes from what the self-similar blob scenario that you're trying to get the smaller, smaller blob to now having a liquid toying machine to get smaller, smaller, smaller, and somehow seeing how that could be used to say something about a blow up. I mean, that's a big leap. So this precedent. I mean, so the thing about mathematics is that it's really good at spotting connections between what you think of what you might think of as completely different problems. But if the mathematical form is the same, you can draw a connection.

所以，从你试图让越来越小的自相似结构转变成现在通过液体玩偶机获得越来越小、更小的结构，并设法观察这如何能够用于描述一个爆炸的情况。这是一个很大的飞跃。不过这种做法有先例。数学的特点在于，它非常擅长发现那些看似完全不同的问题之间的联系。如果它们的数学形式相同，你就可以建立联系。

So there's a lot of previously or what is cellular automata. The most famous of which is Conway's Game of Life. This is infinite to speak grid, and any given time the grid is occupied by a cell or it's empty. And there's a very simple rule that tells you how these cells evolve. So sometimes cells live and some of the day they die. And when I was a student, it was a very popular screen saver to actually just have these animations going on. And they look very chaotic. In fact, they look a little bit like terribly on a float.

所以，有很多以前的研究，或者说是关于元胞自动机的研究。其中最著名的是康威的生命游戏。这是一种无限延展的网格，在任何给定的时间，网格上的格子要么被一个细胞占据，要么是空的。这个游戏有一个非常简单的规则来指引这些细胞的变化。因此，有时候细胞会继续存在，而有时候它们会消亡。当我还是学生的时候，生命游戏作为屏幕保护程序非常流行，因为屏幕上不断播放的动画看起来非常混乱。实际上，这些动画有时看起来就像是漂浮在水面上的杂乱无章的东西。

Sometimes. But at some point, people discovered more and more interesting structures within this Game of Life. So for example, they discovered a single glider. So a glider is a very tiny configuration of four or five cells, which evolves and it just moves at a certain direction. And that's like this vortex rings. So this is an analogy. The Game of Life is kind of a discrete equation. And the fluid Navi-Sok is a continuous equation. But mathematically, they have some similar features.

有时候。但是在某个阶段，人们在这个生命游戏中发现了越来越多有趣的结构。例如，他们发现了一种叫做滑翔机的东西。滑翔机是一种由四五个细胞组成的非常小的配置，它会演变并朝某个方向移动。这就像漩涡环。这是一个类比：生命游戏是一种离散方程，而流体纳维-斯托克斯方程是一个连续方程。但在数学上，它们有一些相似的特征。

And so all the time, people discovered more and more interesting things that you could build within Game of Life. The Game of Life is a very simple system. It only has three or four rules to do it. But you can design all kinds of interesting configurations inside it. There's some called a glider gun that does nothing of spit out gliders one at a time. And then after a lot of effort, people managed to create and gates and all gates for gliders. Like this is massive ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out. If maybe if both of the streams have gliders, then there we are and output stream. But if only one of them does, then nothing comes out.

在这段时间里，人们不断发现在「生命游戏」中可以建造越来越多有趣的东西。「生命游戏」是一个非常简单的系统，只需要三到四个规则就可以操作，但你可以在其中设计各种有趣的结构。例如，有一种叫做滑翔机枪的结构，它只会不断生成滑翔机。而经过大量努力，人们成功地创造出了用于处理滑翔机的「与门」(AND gate) 和「或门」(OR gate)。这是一种非常复杂的结构，如果有两股滑翔机流进来，可能会产生一股滑翔机流出去。如果两股流中都有滑翔机，那么就会有输出，但如果只有一股有滑翔机，则不会有输出。

So they could build something like that. And once you could build and these basic gates, then just from software engineering, you can build almost anything. You can build a touring machine. I mean, it's a kind of enormous steam pump type things. They look ridiculous. But then people also generate self-replicating objects in the Game of Life, a massive machine, a phenomenal machine, which over a huge period of time and they're always glider guns inside doing these very steam pump calculations. They would create another version of itself which could replicate. That's so incredible. A lot of this was like community crowdswashed by like amateur mathematicians actually. So I knew about that work. And so that is part of what inspired me to propose the same thing whenever you're stoked.

他们就能建造那样的东西。一旦你能构建这些基本的门电路，仅凭软件工程，你几乎能够建造任何东西。例如，一个图灵机。我指的是那种巨大的蒸汽泵类型的东西，它们看起来很荒诞。但在人们在生命游戏中也创造了可以自我复制的物体，一种巨大的机器，一种非凡的机器，历经漫长的时间，总是有滑翔机枪在里面进行这些非常蒸汽朋克的计算。这些机器能够创建另一个可以自我复制的版本，非常不可思议。很多这样的成就其实是由业余数学家社区众包出来的。我对这些工作有所了解，这也是为什么我每当情绪高涨时，都会提议类似事情的部分灵感来源。

Now, if you're just a much, as I said, analog is much worse in digital. It's going to be, you can't just directly take the constructions in the Game of Life and pump them in. But again, it shows as possible. There's a kind of emergence that happens with these cellular automata local rules, maybe similar to fluids. I don't know. But local rules operating at scale can create these incredibly complex dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that? The thing is, you can get this emerging very complicated structures, but only with very carefully prepared initial conditions.

这段话可以翻译为如下中文：那么，正如我所提到的，模拟在数字领域的效果要差得多。你不能直接将“生命游戏”中的构造直接应用进来。然而，这再次展示了这种可能性。通过这些元胞自动机的局部规则，会产生一种类似于流体的复杂现象。我不太确定，但局部规则在大规模运作时可以创造出令人难以置信的复杂动态结构。你认为这其中有任何部分适合数学分析吗？我们是否有工具可以对其进行深刻分析？关键在于，你可以得到这种非常复杂的结构，但只有在非常精心准备的初始条件下才会出现。

So these glider guns and gates and sort of machines, if you just plant randomly, some cells, and that you not see any of these. And that's the analogous situation of Navier Stokes again, that with typical initial conditions, you will not have any of this weird computation going on. But basically through engineering, especially designing things in a very special way, you can pick clever constructions. And one of the best possible to prove the sort of the negative of like, basically prove that only through engineering can you ever create something interesting. This is a recurring challenge in mathematics that I call the dichotomy between structure and randomness. That most objects that you can generate in mathematics are random.

所以，这些滑翔机枪、门和类似的机器，如果你只是随机放置一些细胞，就不会看到其中的任何一个。这就类似于纳维-斯托克斯方程的情况，通常的初始条件下，你不会看到这种奇怪的计算发生。但基本上，通过工程手段，特别是以一种非常特殊的方式设计，你可以选择巧妙的构造。最好的证明之一是：只有通过工程设计，你才能创造出有趣的东西。这是数学中一个反复出现的挑战，我称之为结构与随机性的二分法。大多数数学中能生成的对象都是随机的。

They look like random. The digits are pie. Well, we believe there's a good example. But there's a very small number of things that have patterns. But now, you can prove something as a pattern by just constructing something like, if something has a simple pattern and you have a proof that it does some of the repeated itself every so often, you can do that. And you can prove that, for example, you can prove that most sequences of digits have no pattern. So like, if you just pick digits randomly, there's some called low large numbers that tells you you're going to get as many ones as two's in the long run. But we have a lot fewer tools to if I give you a specific pattern like the digits of pi, how can I show that this doesn't have some weird pattern to it?

它们看起来很随机。这些数字是圆周率的数字。我们相信存在一个很好的例子。不过，有非常少的东西具有模式。然而，现在你可以通过构造某些东西来证明某样东西是有模式的。比如，如果某样东西有简单的模式，并且你有证据证明它在某些时候重复出现，你就可以做到这一点。而且你可以证明，例如，你可以证明大多数数字序列是没有模式的。因此，如果你只是随机选择数字，有一个叫做大数法则的理论告诉你，长远来看，你得到的1和2的数量是相近的。但是，如果我给你一个特定的模式，比如圆周率的数字，我们就很缺少工具去证明它没有一些奇怪的模式。

Some other work that I have spent a lot of time on is to prove or construct your theorems or inverse theorems that give tests for when something is very structured. So some functions are what's going to add to it. If I give you a function, I'm asking that natural numbers, the natural numbers. So maybe two maps to four, three maps to six and so forth. Some functions, what's going to additive? Which means that if you add two inputs together, the output gets added as well. For example, a multiply by constant. If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and adding them together.

我花了很多时间做的另一项工作是证明或构建你的定理或逆定理，这些定理可以用来测试某物是否具有非常结构化的特性。其中一些函数是关于要加到其中的内容的。如果我给你一个函数，我指的是自然数，例如2映射到4，3映射到6，等等。有些函数具有可加性，这意味着如果你把两个输入相加，输出结果也会相加。比如，一个常数乘法函数。如果你把一个数字乘以10，那么如果你把a加b，然后乘以10，这相当于分别把a和b都乘以10再相加。

So some functions additive. Some functions are kind of additive, but not completely additive. So for example, if I take a number n, I multiply by the square root of two and I take the integer part of that. So 10 by square root of two is like 14 points, something. So 10, I'm up to 14, 20, I'm up to 28. So in that case, add additive to these two then. So 10 plus 10 is 20 and 14 plus 20 is 28. But because of this rounding, sometimes there's round of errors and sometimes when you add a plus b, this function doesn't quite give you the sum of the two individual outputs, but the sum plus minus one. So it's almost additive, but not quite additive.

有些函数是加法性质的。有些函数有点像加法性质，但并不完全是加法性质的。举个例子，如果我取一个数 n，将其乘以二的平方根，然后取这个结果的整数部分。例如，当 n 是 10 时，乘以平方根后的结果是 14 点多，取整数部分就是 14；当 n 是 20 时，结果是 28。所以在这种情况下，10 和 10 相加是 20，而 14 和 20 相加是 28。不过，由于取整的原因，有时会出现小的误差。当你计算 a 加 b 时，这个函数的结果可能不是两个独立计算结果的和，而是和加上或减去 1。因此，这个函数是几乎具有加法性质，但不完全是。

So there's a lot of useful results in mathematics and I've worked a lot on 12 things like this, to the effect that if a function has to exhibit some structure like this, then it's basically, there's a reason for why it's true and the reason is because there's some other nearby function, which is actually completely structured, which is explaining this sort of partial pattern that you have. And so if you have these little inverse theorems, it creates this sub-decordomy that either the objects that you study are either have no structure at all or they are somehow related to something that is structured. And in either way, in either case, you can make progress.

在数学中，有许多有用的结果。我在这方面做了很多研究，尤其是研究类似的12种情况。如果一个函数必须表现出某种结构，那么这是有原因的。这个原因是因为有另一个附近的函数，它完全有结构性，并可以解释你看到的这种部分模式。所以，如果有这些小的反定理，就会产生这样一个二分法：你研究的对象要么毫无结构，要么与某种有结构的东西有关。不论哪种情况，你都可以有所进展。

A good example of this is that this is old theorem in mathematics called semi-radity theorem proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers, the patterns of arithmetic progression, things like 35 and 7 or 10, 15 and 20. And some really, some really proved that any set of numbers that are sufficiently big, also called positive density, has arithmetic precautions in it of any length you wish. So for example, the odd numbers have a set of density one-half and they contain arithmetic precautions of any length. So in that case, it's obvious because the odd numbers are really with these structures.

这是一个很好的例子，即1970年代证明的一个旧数学定理，称为半稀疏性定理。它涉及在一组数中寻找某种类型的模式，即等差数列的模式，比如35和7，或者10, 15和20之类的数。而且有人确实证明了，只要数集足够大，也称为正密度，那么其中就包含任何长度的等差数列。比如，奇数集的密度是二分之一，并且它们包含任意长度的等差数列。所以在这个例子中，这很明显，因为奇数本身就具有这样的结构。

I can just take 11, 13, 15, 17, I can easily find arithmetic precautions in that set. But there are many of them also applies to random sets. If I take this set of odd numbers and I flip a coin for each number and I only keep the numbers for which I got a heads. So I just flip coins, I just randomly take out half the numbers, I keep one half. So that's the set that has no patterns at all. But just from random fluctuations, you will still get a lot of ethnic progressions in that set. Can you prove that there's arithmetic progressions of arbitrary length within a random...

我可以选择11, 13, 15, 17这个集合，并且很容易在这个集合中找到等差数列。但是，这些特性也适用于许多随机集合。如果我选择这个奇数集合，然后为每个数字抛硬币，只保留那些正面朝上的数字。这样，我就随机去掉了一半的数字，保留另一半。因此，这个集合没有任何规律。但即使是从随机波动中，你仍然会在这个集合中发现很多等差数列。你能证明在随机集合中存在任意长度的等差数列吗？

Yes, I mean, one of the infinite monkey theorem. Usually, mathematicians give boring names to theorists, but occasionally they give colorful names. Yes. The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out text randomly. Almost surely one of them is going to generate the entire school of hamlet or any other finite string of text. It will just take some time, quite a lot of time actually. But if you have an infinite number, then it happens.

是的，我指的是著名的“无限猴子定理”。通常，数学家给定理起的名字都很枯燥，但偶尔也会有一些有趣的名字。这个定理的通俗版是这样的：假设有无限多只猴子，每只猴子面前都有一台打字机，它们随机敲打键盘。几乎可以确定，其中一只猴子会敲出《哈姆雷特》全文或其他任何有限长度的文本。当然，这需要时间，实际上还需要相当长的时间。但如果猴子的数量是无限的，这件事就一定会发生。

So basically, the theorem says that if you take an infinite string of digits or whatever, eventually any finite pattern you wish or you merge, it may take a long time, but it will eventually happen. In particular, ethnic progressions of any length, what eventually happen, a hypodinear, an extremely long random sequence for this to happen? I suppose that's intuitive. It's just infinity. Yeah, infinity absorbs a lot of sense.

基本上，这个定理说明，如果你有一个无限长的数字串或者其他东西，那么任何有限的模式，无论多长时间，最终都会出现。尤其是任何长度的数列，最终都会出现。要让这种情况发生，可能需要一个极其漫长的随机序列。这种说法听起来有点直观，因为这是无穷大。是的，无穷大包含了很多可能性。

Yeah. How are we humans supposed to deal with infinity? Well, you can think of infinity as an abstraction of a finite number of which you do not have a bound for. That, you know, I mean, so nothing in real life is truly infinite. But you know, you can you know, you can ask these old questions like, what about how much money is I wanted? You know, what if I could go as fast as I wanted? And a way in which mathematicians formalize that is, mathematics has found a formalism to idealize instead of something being extremely large, extremely small to actually be exactly infinite or zero.

是啊，我们人类应该如何处理无限呢？你可以把无限看作是一个没有明确界限的有限数字的抽象概念。在现实生活中，没有什么是真正无限的。但是，你可以问一些古老的问题，比如，我想要多少钱呢？或者，如果我能以我想要的速度行驶呢？数学家通过数学形式化的方法，将极大或极小的概念理想化为真正的无限或零。

And often the mathematics becomes a lot cleaner. When you do that, I mean, in physics, we joke about assuming spherical cows, you know, like rule of problems, I've got all kinds of rule of effects, but you can idealize sense of things to infinity, sense of something to zero. And the mathematics becomes a lot simpler to work with it. I wonder how often using infinity forces us to deviate from the physics of reality.

在这种情况下，数学常常变得更加简洁。在物理学中，我们常开玩笑说假设"球形奶牛"，这就像是处理问题的原则一样。我可能遇到各种各样复杂的因素，但可以通过将某些事物理想化得无限大或将某些因素简化为零，使数学变得更容易处理。我在想，使用无穷大的方法究竟有多常会让我们偏离现实物理的本质。

Yeah, so there's a lot of pitfalls. So, you know, we spend a lot of time, you know, undergraduate math classes, teaching analysis, and analysis is often about how to take limits. And whether you know, so for example, A plus B is always B plus A. So when you have a finite number of terms, you add them, you can swap them and there's no problem. But when you have an infinite number of terms, they sort of show games you can play where you can have a series which converges to one value, but you rearrange it and suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity. You have to introduce these epsilons and deltas and there's a certain type of wave of reasoning that helps you avoid mistakes.

是的，这里面有很多隐患。比如，在本科数学课上，我们花了很多时间研究分析，而分析往往涉及如何取极限。我们都知道，比如A加B总是等于B加A。所以当你处理有限项时，你可以随意交换它们的位置，没有问题。但如果是无限项，就可能会出现一些奇妙的情况，比如一个序列收敛到一个值，但如果你重新排列它，就可能会收敛到另一个值。这就是为什么在处理无穷时，你必须非常小心，知道自己在做什么。你需要引入像ε（epsilon）和δ（delta）这样的概念，并运用特定的推理方式来避免出错。

In more recent years, people have started taking results that are true in infinite limits and was sort of finalizing them. So you know, that's something true eventually, but you don't know when, now give me a rate. Okay, so such a thing. If I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do I have to wait for him to come out? And that's a more quantitative question. And this is something that you can attack by purely finite methods and you can use your finite intuition. And in this case, it turns out to be exponential in the length of the text that you're trying to generate. And so this is why you never see the monkeys create hamlet. You can maybe see them create a fuller of wood, but nothing big.

近年来，人们开始关注在无限条件下成立的结论，并尝试将其具体化。也就是说，一些事情在无限情况下最终会成立，但我们不知道具体何时成立，因此我们需要一个速度或比率的概念。举个例子，如果我没有无限多的猴子，而是有大量但有限的猴子，我需要等多久才能让它们随机敲出作品呢？这就成为了一个更为量化的问题。你可以用完全有限的方法来解决这个问题，利用你的有限直觉。结果表明，所需时间与想要生成文本的长度呈指数关系。这就是为什么你几乎不可能看到猴子随机敲出《哈姆雷特》这样的作品，它们也许可以敲出一些较简单的词句，但不可能是复杂的作品。

And so I personally find once you find it high, say infinite statement, it's just a much more intuitive and it's no longer so weird. So even if you're working with infinity, it's good to find it out so that you can have some intuition. Yeah, the downside is that the finite type proves that just much, much messier. And so the infinite ones are found first, usually like decades earlier, and then later on people find it high. So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding of seeing the world? Maybe you can throw an engineering in there. You mentioned your wife is an engineer, give a new perspective on circuits.

所以我个人觉得，一旦你找到了一个高度抽象的观点，比如无限的概念，它就变得更直观，也不再那么令人困惑。即使你在处理无限的事物，找到它们也是好的，这样你就能有一些直觉。缺点是有限种类的证明通常要复杂得多，因此无穷的例子往往是最早被发现的，通常要比有限的例子早几十年。然后人们才会发现复杂的有限情况。既然我们提到了很多数学和物理的内容，那么数学和物理作为学科、作为理解世界的方式有什么区别？也许你可以加入工程学的观点。你提到你妻子是工程师，可以给一个关于电路的新视角。

So this is a different way of looking at the world, given that you've done mathematical physics. So you've worn all the hats. Right. So I think science in general is interaction between three things. There's the real world. There's what we observe over the real world observations. And then our mental models as to how we think the world works. So we can't directly access reality. Okay. All we have are the observations which are incomplete and they have errors. And there are many, many cases where we would want to know because of what is the weather like tomorrow. We don't have the observation and we'd like to predict. And then we have these simplified models sometimes making unrealistic assumptions. It's spherical cow type things.

这是一种看待世界的不同方式，尤其是对你这样有数学物理背景的人而言。你在科学领域几乎都涉猎过。我认为科学总体上是三个方面的互动：真实的世界、我们对真实世界的观察，以及我们脑海中关于世界如何运作的模型。我们无法直接接触到现实，能依赖的只有那些不完整且有误差的观察。在很多情况下，我们希望得知一些信息，比如明天的天气如何，却缺乏观察数据，只能依靠预测。而我们的预测模型往往是简化的，有时还会基于一些不切实际的假设，比如“理想球形奶牛”的假设。

Those are the mathematical models. Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations. What mathematics does it, we stay within the model and ask what are the consequences of that model? What observations, what predictions would the model make of the future observations? Or past observations to fit observed data. So there's definitely a symbiosis. I guess mathematics is unusual among other disciplines is that we start from hypotheses like the axioms of a model and ask what conclusions come up from that model.

这些是数学模型。数学关注的是这些模型。科学收集观察结果，并提出可能解释这些观察的模型。数学的工作是留在模型内，探讨该模型的后果是什么？该模型对未来的观察会有哪些预测？或者过去的观察是否能符合已观测的数据。因此，这两者之间确实存在一种共生关系。我想，数学在其他学科中是独特的，因为我们从假设（如模型的公理）开始，探讨从该模型中能得出哪些结论。

In almost any other discipline, you start with conclusions. I want to do this. I want to board a bridge. I want to make money. I want to do this. Then you find the path to get there. There's a lot less sort of speculation about it. Suppose I did this. What would happen? Planning and modeling. Specular fiction maybe is one other place. But that's what I did. Actually, most of the things we do in life is conclusions driven, including physics and science. They want to know where is this asteroid going to go. What is the weather going to be tomorrow?

在几乎任何其他领域，你从结论开始。我想做这个。我想上桥。我想赚钱。我想做这个。然后你会找到实现目标的路径。这其中涉及的推测相对较少。假设我这么做了，会发生什么？也许在计划和建模中有一些这种情况。在虚构小说中可能也是如此。但这就是我所做的。实际上，我们生活中大部分事情都是以结论为导向的，包括物理和科学。他们想知道这个小行星会去哪里。明天天气会怎么样。

But physics also has this other direction of going from the axioms. What do you think there is this tension in physics between theory and experiment? What do you think is the more powerful way of discovering truly novel ideas about reality? Well, you need both. Top down on bottom up. It's a really an interaction with you in all these things. Over time, the observations and the theory and the modeling should both get closer to reality. Initially, this is the case. They're always far apart to begin with. But you need one to figure out where to push the other. If your model is predicting anomalies that are not picked up by experiment, that tells the experimenters where to look to find more data to refine the models.

这句话的意思是：在物理学中，除了从公理出发的研究方向之外，还有另一种方法。你认为理论和实验之间为什么在物理学中存在这种紧张关系？你觉得哪种方法更能有效地发现有关现实的全新想法？其实，两者都需要。自上而下和自下而上都是相辅相成的过程。随着时间的推移，观察和理论建模应该逐渐更接近现实。一开始的时候，它们常常相距甚远。但你需要一种方法来推动另一种方法的发展。如果你的模型预测出实验无法观测到的异常现象，这就可以指引实验人员去寻找更多的数据，从而优化模型。

It goes back and forth. Within mathematics itself, there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely 99% of mathematics. It's theoretical mathematics. There's a very tiny amount of experimental mathematics. I mean, people do do it. If they want to study prime numbers or whatever, they can generate large data sets. Once we had a computer, we'd be able to do it a little bit.

这件事情有来有回。在数学本身中，也包括理论和实验的部分。只是直到最近，理论在数学中几乎完全占据主导地位，占到了99%。这种数学属于理论数学。而实验数学则非常少。我的意思是，确实有人在做实验数学。如果有人想研究素数或者其他什么，他们可以生成大型数据集。有了计算机，我们就能稍微进行一些实验数学。

Although, even before, like Gals, for example, he discovered the most basic theory in number theory to call the prime number theorem, which predicts how many primes that have to a million, have to a trillion. It's not an obvious question. Basically, what he did was he computed most of these, by himself, but also hired human computers, people whose professional job it was to do arithmetic. To compute the first 100,000 primes or something and made tables and made a prediction.

即使在此之前，比如说伽尔斯，他发现了数论中最基本的理论，称为素数定理。这个定理可以预测在一百万以内、一万亿以内有多少个素数。这不是一个显而易见的问题。基本上，他自己计算了大部分这些数据，但同时也雇用了专业的计算人员，这些人的工作就是进行算术计算。他们计算了前十万个素数或者类似的数据，制作了表格，并进行了预测。

That was an early example of experimental mathematics. Very recently, it was not... Theoretical mathematics was just much more successful. Doing complicated mathematical computations was just not feasible until very recently. And even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically. There's something called the combinatorial explosion. If you want to study, for example, Zermadee's theory, you want to study all possible subsets of numbers 1 to 1000.

这算是一个早期的实验性数学的例子。直到最近，理论数学一直更为成功，因为进行复杂的数学计算长期以来都很困难，直到最近才变得可行。即使到今天，尽管我们有强大的计算机，仍然只有一些数学问题可以通过数值方法进行探索。这是因为存在一种叫做“组合爆炸”的问题。比如说，如果你想研究Zermadee的理论，你需要研究数字1到1000的所有可能子集。这会导致计算量急剧增加，难以用计算机完全解决。

There's only 1000 numbers. How bad could it be? It turns out the number of different subsets of 1 to 1000 is 2 to the power 1000, which is way bigger than any computer can currently continue. Any computer ever will have a computer in your brain. There are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example. The number of chess positions we can't get a computer to fully explore.

只有1000个数字。这能有多复杂呢？然而，实际上，从1到1000这1000个数字中可以构成的不同子集数量是2的1000次方，这个数字远远大于任何现今的电脑可以处理的范围。即便将来有再强大的电脑，也不可能装进你的大脑。有些数学问题很快就变得无法用直接的暴力计算法解决。国际象棋就是一个著名的例子。我们无法让电脑完全探索所有可能的棋局。

But now we have AI. We have tools to explore this space, not with 100% guarantees of success, but with experiment. So we can empirically solve chess now. Very, very good AI is that they don't explore every single position in the game tree, but they have found some very good approximation. And people are using actually these chess engines to do experimental chess. They're revisiting all chess theories about, oh, this type of opening, this is a good type of movement.

但是现在我们有了人工智能。我们有工具可以探索这个领域，虽然不能百分之百保证成功，但可以通过实验来进行探索。因此，我们现在可以通过经验来解决国际象棋的问题。非常优秀的人工智能并不是探索游戏树中的每一个位置，而是找到了非常好的近似解。人们实际上正在使用这些国际象棋引擎进行实验性的国际象棋研究，他们正在重新审视所有的象棋理论，比如，这种开局方式，这是一种好的走法。

This is not. And they can use these chess engines to actually refine in some case overturn conventional wisdom about chess. And I do hope that mathematics will have a larger experimental building in the future perhaps powered by AI. Well, of course, talk about that. In the case of chess, and there's a similar thing in mathematics, I don't believe it's providing a kind of formal explanation of the different positions. But it's just saying which position is better, not that you can intuit as a human being.

这并不如此。在某些情况下，人们可以使用这些国际象棋引擎来改进甚至推翻传统的象棋智慧。我希望未来数学能够有更大的实验性发展，也许由人工智能驱动。当然，我们可以讨论这个话题。在象棋中，数学中也有类似的情况，我不认为这些工具能为不同的棋局提供一种正式的解释。它们只是指出哪个棋局更优，而不是人类可以直观理解的。

And then from that, we humans can construct a theory of the matter. You've mentioned the Plato's cave algorithm. So it gives people to know it's where people are observing shadows of reality, not reality itself. And they believe what they're observing to be reality. Is that in some sense what mathematicians and maybe all humans are doing is looking at shadows of reality? Is it possible for us to truly access reality?

然后，根据这些观察，我们人类可以构建一个关于事物的理论。你提到了柏拉图洞穴的比喻。这个比喻告诉我们，人们观察到的是现实的影子，而不是真实的本身，他们却误以为自己看到的就是现实。从某种意义上来说，数学家，甚至所有人类，是不是也只是在观察现实的影子呢？我们有可能真正接触到真实的世界吗？

Well, there are these three ontological things. There's actual reality, there's observations and models. And technically, they are distinct. And I think they will always be distinct. But they can get closer over time. And the process of getting closer often means that you have to discard your initial intuitions. Astronomy provides great examples. An initial model of the world is flat because it looks flat.

好的，有三个本体上的事物：实际现实、观察和模型。从技术上讲，它们是不同的，我认为它们将始终是不同的。但是，它们可以随着时间的推移变得更加接近。而这种接近的过程通常意味着你需要抛弃最初的直觉。天文学就提供了很好的例子。最初对世界的模型是平的，因为看上去确实是平的。

And it's big. And the rest of the universe, the sky is not like the sun, for example, looks really tiny. And so you start off with a model which is really far from reality. But it fits kind of the observations that you have. So things look good. But over time, as you make more and more observations, bring it closer to reality, the model gets dragged along with it. And so over time, we had to realize that the Earth was round, that it spins.

它很大。而宇宙的其他部分，比如天空，并不像太阳那样庞大，看起来真的很小。因此，你一开始用的是一个与现实相去甚远的模型。但这个模型能够大致匹配你已有的观察结果，所以一开始看起来不错。但随着时间的推移，当你进行越来越多的观察，使其更接近现实时，这个模型也被一点一点带向真实。因此，我们逐渐认识到地球是圆的，而且它会自转。

It goes around the solar system. So it goes on the galaxy and so on and so forth. And the guys part of the universe, humans are expanding. Expansions are self-expanding, accelerating. And in fact, very recently, in this year or so, even the evolution of the universe still is this evidence that is non-constant. And the explanation behind why that is, it's catching up. It's catching up.

它围绕着太阳系运转。因此，它也在银河系中运转，如此不断扩展。而在人类扩展的宇宙中，扩张是自我扩展的，并且在加速。事实上，就在最近的这一年左右，宇宙的演化仍然显示出了变化的迹象，而不是恒定的。至于为什么会这样，相关的解释正在逐步得到理解和追赶。

I mean, it's still the dark matter, dark energy. We have a model that sort of explains that fits the data really well. It just has a few parameters that you have to specify. But so people say all that spud factors, with enough spud factors, you can explain anything. But the mathematical point of the model is that you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10-up to observations, that is completely useless model. And so it's got all the fitted. But if you have a model with two parameters and it explains a trillion observations, which is basically, so the dark matter model, I think it has like 14 parameters, and it explains petabytes of data that the astronomer's have. You can think of all the theory.

我的意思是，暗物质和暗能量依然是我们关注的重点。我们有一个模型，它能很好地解释并适配这些数据。这个模型只需要你指定几个参数。然而，有人可能会说，你只要有足够多的参数就可以解释任何事情。但从数学的角度来看，模型的关键是要让模型的参数数量比观测数据点更少。如果你的模型有10个参数却只能解释10个观测数据点，那这个模型是没有价值的。然而，如果你的模型只有两个参数但能解释一万亿个观测数据点，那就是一个非常有效的模型。暗物质模型大概有14个参数，而它能解释天文学家所拥有的大量数据。你可以将其视为一种理论。

One way to think about physical mathematical theory is a compression of the universe and a data compression. So you have these petabytes of observations. You'd like to compress it to a model, which you can describe in five pages and specify a certain number of parameters and it can fit to reasonable accuracy, almost all of the observations. The more compression that you make, the better your theory. In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all. It's the unreasonable effect and it's the mathematics. I'm not a quote like that. The most incompressible thing about the universe is that it is comprehensible. And not just comprehensible. You can do an equation like E equals empty squared. There is actually some mathematical possible explanation for that. This is phenomenal in mathematical universality.

一种理解物理数学理论的方法是将其视为对宇宙的一种压缩和数据压缩。你拥有大量的观测数据，希望将其简化成一个能够用五页纸描述的模型，并用一定数量的参数来表述，这个模型可以以合理的精度适用于几乎所有的观测结果。理论压缩得越多，就越好。事实上，宇宙及其中一切的一个巨大惊奇是它竟然能够被压缩。这也是数学的神奇之处。正如某句话所说，宇宙中最不可压缩的事情就是它是可以被理解的。而且不仅是可以被理解的，你还可以写出像E=mc²这样的公式，并且实际上能够找到一些数学解释。这是数学普遍性的一种奇妙现象。

Many complex systems at the macro scale are coming out of lots of tiny new interactions at the macro scale. Normally because of the common form of explosion, you would think that the macro scale equations must be infinitely exponentially more complicated than the macro scale ones. They are, if you want to solve them completely exactly. If you want to model all the atoms in a box of air, I have like algebra's numbers, you're monkeys. There's a huge number of particles. If you actually have to track each one, it will be ridiculous. But certain laws emerge at the macro scale that almost don't depend on what's going on at the macro scale or only depend on a very small number of parameters.

在宏观层面，许多复杂系统是由许多微小的新互动形成的。通常，由于爆炸式增长的常见形式，你可能会认为宏观层面的方程比宏观层面的方程要复杂无数倍。实际上，如果你想完全精确地解决这些方程，那的确是这样。如果你想要模拟一个空气盒子里的所有原子，那就像要处理大量数字一样，简直就是天方夜谭。跟踪每个粒子简直是不可能的。然而，一些自然规律在宏观层面显现出来，它们几乎不依赖于微观层面的情况，或者只依赖于少数几个参数。

So if you want to model a gas of, you know, quintillion particles in a box, you just need to know temperature and pressure and volume in a few parameters, like 506. And it models almost everything you need to know about these 10th or 23 or whatever particles. So we don't understand universality anywhere new as we would like mathematically. But there are much simpler toy models where we do have a good understanding of why universality occurs. The most basic one is the central element theorem that explains why the bell curve shows up everywhere in nature. So many things are distributed by what's got a Gaussian distribution, a famous bell curve. There's now even a meme with this curve. And even the meme applies broadly. The universality to the meme.

如果你想在一个盒子里模拟包含万亿级（例如10的23次方）粒子的气体，只需了解温度、压力和体积，还有几个参数（比如506），就能模拟出关于这些粒子的绝大多数信息。我们在数学上还没有达到对普适性（universality）的深入理解。不过，有一些更简单的模型可以帮助我们很好地理解为什么普适性会发生。最基本的模型就是中心极限定理，它解释了为什么正态分布的钟形曲线在自然界中无处不在。许多事物都遵循高斯分布（著名的钟形曲线），甚至还有一个关于这个曲线的网络梗，而且这个梗也可以广泛地应用于不同领域。这个现象的普适性甚至在网络梗中也得到了体现。

Yes, you can go matter if you like. But there are many, many processes. For example, you can take lots of independent random variables and average them together in various ways. You can take a simple average or more complicated average. And we can prove in various cases that these bell curves, these Gaussian's emerge. And it is a satisfying explanation. Sometimes they don't. So if you have many different inputs and they will correlate it in some systemic way, then you can get something very far from a bell curve show up. And this is also important to know when the situation is really fails. So universality is not a 100% reliable thing to rely on. That global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of Gaussian type behavior.

是的，如果你愿意，你可以更深入地探讨这个问题。但是，这其中涉及到许多、许多的过程。例如，你可以将许多独立的随机变量以各种方式进行平均。你可以采用简单的平均或更复杂的平均方法。我们可以证明，在某些情况下，这些钟形曲线，即高斯分布，会出现。这通常是一个令人满意的解释。但有时情况并非如此。如果你有许多不同的输入，它们以某种系统性的方式相关，那么可能会出现与钟形曲线大相径庭的结果。这也是一个需要注意的点，因为普遍性并不是百分之百可靠的。全球金融危机就是一个著名的例子。人们曾以为抵押贷款违约具有某种高斯类型的行为。

That if you ask if a population of 100,000 Americans with mortgages, ask what proportion of the mortgage is. If everything was decarolated, it would be an ass bell curve. And you can manage risk of options and derivatives and so forth. And it is a very beautiful theory. But if there are systemic shocks in the economy that can push everybody default at the same time, that's very non-gasting behavior. And this wasn't fully accounted for in 2008. Now I think there's some more awareness this is systemic risk is a key up a much bigger issue. And just because the model is pretty and nice, it may not match reality. So the mathematics of working out what models do is really important.

如果你询问10万名拥有房贷的美国人，问他们房贷占的比例是多少。假设一切都是去相关化的，那么分布会呈现一个对称的钟形曲线。这样你就可以管理期权和衍生品等风险，这其中包含一个非常美的理论。然而，如果经济中出现系统性冲击，导致所有人同时违约，那就是一种非常非典型的行为。这一点在2008年的金融危机中并没有被充分考虑到。现在我认为大家对系统性风险的认识提高了很多，意识到这是一个更大的问题。虽然模型看起来完美漂亮，但可能并不符合现实。因此，研究模型的数学运算真的非常重要。

But also the size of validating when the models fit reality and when they don't. I mean, that you need both. And but mathematics can help because it can, for example, the central limit thing was it tells you that if you have certain axioms like non-correlation, that if all the inputs were not correlated to each other, then you have these Gaussian behaviors that things are fine. It tells you where to look for weaknesses in the model. So if you have a method, an understanding of central limit to and someone proposes to use these Gaussian couperlors or whatever to model, deport risk, if you're mathematically trained, you would say, okay, but what are the systemic correlation between all your inputs? And so then you can ask the economists how much risk is that? And then you can you can you can look for that.

同时，还需要验证模型何时符合现实，何时不符合。我的意思是，你需要两者兼顾。不过，数学可以提供帮助。比如，中心极限定理告诉我们，如果你有某些公理，比如非相关性，也就是说所有输入彼此不相关，那么就会出现高斯行为，这时情况是良好的。这个定理还会指引你寻找模型中的弱点。因此，如果你具备中心极限定理的知识，而有人提议使用高斯协同过程来建模，分析风险，如果你有数学训练背景，你会询问：所有输入之间的系统性相关性是多少？然后你可以向经济学家询问这会带来多大的风险，从而进一步探讨问题。

So there's always this synergy between science and mathematics. A little bit on the topic of universality. You're known and celebrated for working across an incredible breadth of mathematics. So I'm an instant of Hilbert a century ago. In fact, the great fields metal winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert. He's a colleague of yours. But anyway, so you are known for this ability to go both deep and broad in mathematics. So you're the perfect person to ask, do you think there are threads that connect all the disparate areas of mathematics?

科学与数学之间总是存在着一种协同作用。在普适性这个话题上，您以横跨广泛数学领域而闻名并受到赞誉。我想到了一百年前的希尔伯特。事实上，伟大的菲尔兹奖得主数学家蒂姆·高尔斯曾说，您是我们所能见到的最接近希尔伯特的人。蒂姆是您的同事。不管怎样，您以在数学中既能深入又能广泛贯通而闻名。因此，您是个完美的人选来回答这个问题：您认为是否有连接所有不同数学领域的线索？

Is there kind of deep underlying structure to all of mathematics? This certainly a lot of connecting threads and a lot of the progress of mathematics has can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. So in the times of ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both. You could work both on geometry most famously, but also on numbers. But they were not really considered related. I mean, a little bit like, you could say that this length was five times this length because you could take five copies of this length and so forth.

是否有一种深层的基础结构贯穿于整个数学？毫无疑问，数学中有许多相互关联的线索，而数学的发展在很大程度上可以用两个曾经没有关联的数学领域之间找到联系的故事来呈现。一个古老的例子是几何学和数论。在古希腊时期，这两者被视为不同的学科。也就是说，数学家们同时研究几何学和数论，他们可能在几何学上享有盛名，但同时也研究数字。不过，它们并未被真正视为相互关联的领域。就像在度量时，你可以说一个长度是另一个长度的五倍，因为你可以将这个长度复制五次，依此类推。

But it wasn't until Descartes, you've really realized that you could develop the geometry that you can parameterize the plane a geometric object by two real numbers at every point. And so geometric problems can be turned into problems about numbers. And today, this feels almost trivial. There's no content to list. Of course, the plane is XX and Y. Because that's what we teach and it's internalized. But it was an important development that these two fields were unified. And this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated and now we have a suitable algebraic geometry that connects them and over and over again.

直到笛卡尔，人们才真正意识到可以通过在平面上的每一个点用两个实数来参数化几何对象。这使得几何问题可以转化为数字问题。今天，这几乎显得微不足道，平面自然就是由X和Y轴构成的。这是我们教授的内容，已深入人心。然而，将代数与几何统一起来是一个重要的发展。这样的过程在数学中一再重复。代数和几何曾经是分开的领域，而现在我们有了将它们连接起来的合适的代数几何学，并不断在这方面取得进展。

And that's certainly the type of mathematics that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think hedgehogs and fox. Fox knows many things a little bit, but a hedgehog knows one thing very, very well. And in mathematics, there's definitely both hedgehogs and foxes. And then there's people who are kind of who can play both roles. And I think I do a collaboration between mathematicians involves very, you need some diversity. Fox working with many hedgehogs all vice versa. So yeah, but I identify mostly as a fox certainly. I like arbitrage somehow, like learning how one field works, learning the tricks of that wheel and then going to another field, which people don't think it is related, but I can adapt the tricks.

这正是我最喜欢的数学类型。我认为作为一个数学家有不同的风格。我喜欢用“刺猬和狐狸”来比喻：狐狸知道很多事情，但懂得都不深，而刺猬则对某一件事了解得非常透彻。在数学界，确实有像刺猬和狐狸一样的数学家，还有一些人能够兼具这两种风格。我认为数学家的合作需要多样性，比如狐狸与许多刺猬的合作，或反过来。而我大多数情况下更像是一只狐狸。我喜欢类似套利的过程，学习一个领域的运作方式，掌握其中的技巧，然后应用到另一个通常不被认为相关的领域。

So see the connections between the fields. So there are other mathematicians who are far deeper than I am. They're really hedgehogs. They know everything about one field and they're much faster and more effective in that field. But I can I can give them these extra tools. I mean, you said that you can be both a hedgehog and the fox depending on the context and depending on the collaboration. So what can you if it's at all possible speak to the difference between those two ways of thinking about a problem, say you're encountering a new problem, you know, searching for the connections versus like very singular focus.

所以，要看到各个领域之间的联系。有些数学家比我更有深度，他们真正是一种“刺猬型”人物。他们对某个领域无所不知，在那个领域中更快更有效。但我能够为他们提供额外的工具。你提到过，根据不同的背景和合作，个人可以既是“刺猬”也是“狐狸”。那么，能否谈一下这两种思考问题方式的区别呢？比如，当你遇到一个新问题时，是寻找联系，还是专注于一个点？

I'm much more comfortable with the fox paradigm. Yeah, so yeah, I like looking for analogies, narratives. I spend a lot of time. If it is a result, I see it in one field and I like the result. It's a cool result, but I don't like the proof. Like it uses types of mathematics that I'm not super familiar with. I often try to re-prove it myself using the tools that I favor. Often my proof is worse, but by the exercise you're doing so, I can say, oh, now I can see what the other proof was trying to do. And from that, I can get some understanding of the tools that I have used in that field.

我更习惯于“狐狸范式”。是的，我喜欢寻找类比和叙述。我花很多时间在这上面。如果我在某个领域看到一个结果，并且我喜欢这个结果，但不喜欢它的证明过程，因为它用到了我不太熟悉的数学类型。我常常尝试用自己喜欢的方法重新证明它。尽管我自己的证明通常不如原来好，但通过这个练习，我可以理解原来的证明想要做什么。这样，我可以对自己在该领域使用的工具有更好的理解。

So it's very exploratory, very doing crazy things and crazy fields and like reinventing the wheel a lot. Whereas the hedgehog style is, I think much more scholarly, you know, you're very knowledge based. You stay up to speed on like all the developments in this field, you know, all the history. You have a very good understanding of exactly the strengths and weaknesses of each particular technique. I think you'd rely a lot more on calculation than sort of trying to find narratives. So yeah, I mean, I could do that too, but the other work extremely good at that.

这是一种非常探索性的风格，涉及到做一些疯狂的事情和尝试不一样的领域，经常是不断重新发明轮子。而刺猬型风格，我认为更像是学术型的，你会非常注重知识，保持对这个领域所有进展的了解，掌握所有的历史。你对每种特定技术的优缺点都有非常好的理解。我认为你会更加依赖计算，而不是尝试去寻找故事。所以，是的，我也可以做到这一点，但另一个工作在这方面非常出色。

Let's start back and maybe look at a bit of a romanticized version of mathematics. So I think you've said that early on in your life, math was more like a puzzle solving activity when you were young. When did you first encounter a problem or proof where you realize math can have a kind of elegance and beauty to it? That's a good question. When I came to graduate school in Princeton, so John Conway was there at the time. He passed away a few years ago.

让我们回到最初，也许我们可以看一下数学的浪漫化版本。我记得你曾说过，在你生命的早期，数学更像是一种解谜活动。当你年轻时，什么时候你第一次遇到一个问题或证明，让你意识到数学也可以具有一种优雅和美呢？这是个很好的问题。当我来到普林斯顿读研究生的时候，约翰·康威正好在那儿。他在几年前去世了。

But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. So Conway had just said this is an amazing way of thinking about all kinds of things in a way that you would normally think of. So he thought of proofs themselves as occupying some sort of space. So if you want to prove something, let's say that there's infinitely many primes. You're all different proofs. But you could rank them in different axes. Some proofs are elegant, some are long, some proofs are elementary and so forth. So this is cloud.

我记得我去听的第一场研究报告之一是康威的演讲，他所讲的内容被称为极端证明。康威提到这是一种惊人的思维方式，可以应用于各种通常思考事物的方法。他认为证明本身就像占据了一种空间。如果你想证明某件事，比如说证明素数有无限多个，那么你就会有各种不同的证明方法。而这些证明可以按照不同的维度来排序。有些证明优雅，有些长，有些则比较基础等等。这就是所谓的"云"。

The space of all proofs itself has some sort of shape. So he was interested in extreme points of this shape. All these proofs, what is one of these shortest at the expense of everything else or the most elementary or whatever. And so he gave some examples of well-known theorems and then he would give what he thought was the extreme proof in these different aspects. I just found out really eye opening that it's not just getting a proof for a result, it was interesting. But once you have that proof trying to optimize it in various ways, that proofing itself had some craftsmanship to it.

所有证明的集合本身具有某种形状。他对此形状的极端点很感兴趣。这些证明中，是否有某个证明在某种方面最简洁，或者是最基础的。他列举了一些知名定理，并给出了在不同方面上他认为的极端证明。我发现这非常启发人，不仅仅是为了得到一个结果的证明而已，尝试以各种方式优化该证明，这本身也是一种工艺。

It's something for my writing style that when you do your math assignments and undergraduate, your homework and so forth, you're encouraged to just write down any proof that works. As long as it gets a tick mark, you move on. But if you want your results to actually be influential and be read by people, it can't just be correct. It should also be a pleasure to read, motivated, be adaptable to generalize other things. It's the same in many other disciplines, like coding. There's a lot of analogies between math and coding.

在我的写作风格中，有这样一种体会：在大学做数学作业时，你被鼓励只要写出能证明问题的答案即可。只要能得分就算过关。但如果你希望自己的研究结果真正产生影响并被他人阅读，它不仅仅需要正确，还要让人读起来愉悦、有动机，并且能够推广到其他领域。许多其他学科也是如此，比如编程。在数学和编程之间有很多相似之处。

I like analogies if you haven't noticed. But you can code something spaghetti code that works for a certain task and it's quick and dirty and it works. But there's lots of good principles for writing code well so that Alipuk can use it, board upon it, and so on. It has fewer bugs and whatever. There's some of the things with mathematics. First of all, there's so many beautiful things there. And Kamu is one of the great minds in mathematics ever and computer science. Just even considering the space of proofs.

如果你还没注意到的话，我喜欢用类比。你可以写出一段杂乱无章的代码，它虽然很简陋但能够完成特定任务。然而，为了让代码更好、更易于他人使用和改进，有很多编写代码的良好实践。这种代码更少出错，质量更高。在数学领域也有类似的情况。首先，数学本身就充满了许多美丽的事物。而Kamu则是数学和计算机科学领域的一位伟大人物，即便是在探索证明的空间方面，也是如此。

And saying, okay, what does the space look like? And what are the extremes? Like you mentioned coding is an analogies interesting because there's also this activity called the code golf. Oh yeah, yeah, which I also find beautiful and fun where people use different programming languages to try to write the shortest possible program that accomplishes a particular task. Then I believe there's even competitions on this.

你提到编程，这让我想起了一个有趣的活动，叫做“代码高尔夫”。哦对对，我也觉得这个活动既美妙又有趣。在这个活动中，人们用不同的编程语言来尝试写出最短的程序来完成特定任务。我相信这个活动甚至还有比赛。

Yeah, yeah. And it's also a nice way to stress tests, not just the sort of the programs or in this case the proofs but also the different languages. Maybe that's a different notation or whatever to use to tell a comprehensive different task. Yeah, you learn a lot. I mean, it may seem like a frivolous exercise, but it can generate all these insights which if you didn't have this artificial objective to pursue, you might not see.

是的，是的。这也是个很好的方式来进行压力测试，不仅是针对程序或在这种情况下的证明，还可以测试不同的语言。或许是为了使用不同的符号或其他方式来表达复杂的不同任务。是的，你会学到很多东西。我是说，这可能看起来像是一种轻松的练习，但它能带来许多启发，如果你没有这个人为设定的目标去追求，你可能就看不到这些。

What to use the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in beauty is this simplicity. So if you look at E equals obviously squared. So when a few concepts come together, that's why the oil or identity is often considered the most beautiful equation in mathematics. Do you find beauty in that one and the oil identity?

在数学中，什么被认为是最美或最优雅的方程？我指的是，人们在看待美的时候，通常会关注它的简单性。例如，当几个概念融合在一起时，就像著名的爱因斯坦质能方程（E=mc²）那样，这就是为什么欧拉恒等式常被视为数学中最美的公式之一。你觉得这个公式和欧拉恒等式美吗？

Yeah, well, as I said, I mean, what I find most appealing is connections between different things that you like. So if you eat the pie I equals minus one. So yeah, people are always, this is all the fundamental constants. Okay, that's cute. But to me, so the exponential function was interesting. To measure exponential growth. So the compound interest or decay or anything which is continuously growing continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function. Whereas pie comes around from circles. and rotation. If you want to rotate a needle, for example, 100 degrees, you need to rotate by pie radians. And I, complex numbers, represents this hoping should be an imaginary axis of a negative rotation. So a change in direction.

好的，就像我说过的，我最感兴趣的是不同事物之间的联系。如果你吃π，我等于负一。对，人们总是说这些是基本常数。好的，这很有趣。但对我来说，指数函数更有趣。它用于测量指数增长。比如复利增长或衰减，或者任何持续增长或持续减小的东西，这些都可以用指数函数来建模。而π与圆和旋转有关。例如，如果你想旋转一个针100度，你需要旋转π弧度。而i，复数，表示在一个负方向上的虚轴，希望是一个方向的改变。

So the exponential function represents growth and decay in the direction where you really are. When you stick an eye in the exponential, instead of motion in the same direction as your composition, it's a motion as a right angle to your composition. So rotation. And then so if the pie I equals minus one tells you that if you rotate for a time pi, you end up at the other direction. So it unifies geometry through dilation and exponential growth dynamics through this act of classification, rotation by by by. So it connects together all these two mathematics, yeah, that's a question of complex and complex and complex numbers. They will consider almost their own next door neighbors in mathematics because of this identity.

指数函数表示增长和衰减的方向就是你真正所在的方向。当在指数函数中引入一个虚数单位\(i\)时，运动不再沿着原来组成的方向，而是转向与该方向垂直的方向，因此会产生旋转。如果\(\pi i\)等于-1，这意味着如果你旋转\(\pi\)时间，你会到达相反的方向。通过这种分类、旋转的行为，几何通过扩张和指数增长动态得到了统一。这个过程连接了两种数学概念，这是复数在数学中被认为几乎是邻居的原因，因为这种恒等式让它们紧密联系在了一起。

Do you think the thing you mentioned is cute? The collision of notations from these disparate fields is just a frivolous side effect or do you think there is legitimate value in one notation? Although our old friends come together at night. Well, it's confirmation that you have the right concepts. So when you first study anything, you have to measure things and give them names. And initially, sometimes because your model is again too far off from reality, you give the wrong things the best names and you only find out later what's really important. Physicists can do this sometimes. I mean, but it turns out okay.

你认为你提到的东西很可爱吗？这些来自不同领域的符号碰撞仅仅是一个无关紧要的副产品，还是你认为某种符号确实有其价值？尽管我们的老朋友们晚上聚在一起，这也证明了你掌握了正确的概念。因此，当你第一次学习任何东西时，你必须测量它们并给它们命名。起初，由于你的模型与现实相差甚远，可能会错误地将好的名字赋予不重要的东西，后来才发现什么是真正重要的。物理学家有时也会这样做，但结果通常会不错。

So actually, physics of so equals n times squared. Okay, one of the big things was the E. So when Aristotle first came up with his laws of motion and then then Galileo and Newton and so forth, they saw the things they could measure. They could measure mass and acceleration and force and so forth. Newtonian mechanics, for example, ethical is MA, it was the famous Newton's second law of motion. So those were the primary objects. So they gave them the central building in the theory. It was only later after people started analyzing these equations that they always seem to be these quantities that were conserved.

据说，物理学中的某些公式可以用 n 倍于平方的方式表示。一个重要的因素是能量 (E)。当亚里士多德首次提出他的运动定律，然后伽利略和牛顿等人接着研究时，他们观察到一些可被测量的事物，如质量、加速度和力等等。例如，牛顿力学中的 F=ma 就是著名的牛顿第二运动定律。因此，这些是研究的主要对象，成为理论的核心组成部分。后来，人们开始分析这些方程式时，发现总有一些守恒量出现。

So a particular momentum in energy. And it's not obvious that things happen energy. Like it's not something you can directly measure the same way you can measure mass and velocity so forth. But over time, people realized that this was actually a really fundamental concept. Hamilton, eventually in 19th century, reformulated Newton's laws of physics into what it's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object. Once you know how to measure the Hamiltonian of any system, you can just completely detect the dynamics like what happens to it or to all the states like it.

在能量领域有一个特定的动量。事情在能量中发生并不是显而易见的，因为你不能像直接测量质量和速度那样直接测量能量。但是，随着时间的推移，人们意识到这是一个非常基本的概念。19世纪时，哈密顿重新制定了牛顿的物理定律，将其发展为哈密顿力学，其中能量现在被称为哈密顿量，成为了核心对象。一旦你知道如何测量任何系统的哈密顿量，你就可以完全检测到它的动态，比如它会发生什么，或者所有类似状态的情况。

It really was a central actor, which was not obvious initially. And this helped actually, this change of perspective really helped when quantum mechanics came along. Because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt in Newtonian thinking because the other thing was particle and so forth to quantum mechanics. Because I think people was a way, but it just looked really, really weird. Like what is the quantum version of F equals M A? And it's really, really hard to give an answer to that.

最初并不明显的是，它的确是一个核心因素。而从这个角度的改变实际上帮助了量子力学的发展。因为早期研究量子力学的物理学家在试图从牛顿力学的思维方式过渡到量子力学时遇到了很多困难。因为牛顿力学中的概念，比如粒子等，很难融入到量子力学中。量子力学让人感觉很奇怪，比如说F等于MA在量子力学中应该是什么样的？对此问题给出答案是非常困难的。

But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics also is the key object in quantum mechanics that there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than a function, but but again, once you specify it, you specify the entire dynamics. So the circle shown this equation that tells you exactly how quantum systems evolve once you have the Hamiltonian. So side by side, they look completely different objects, you know, like one in those particles, one of those waves and so forth.

事实证明，哈密顿量（Hamiltonian）不仅在经典力学中暗中发挥着重要作用，也是量子力学中的关键对象。在量子力学中，也有一个称为哈密顿量的对象。虽然它的类型不同，被称为算符而不是函数，但一旦你确定了这个对象，你就可以描述整个系统的动态。这就像是一个圆圈，展示了当你有了哈密顿量之后，能够精确地描述量子系统是如何演化的。虽然在外观上，它们可能完全不同，比如一个涉及粒子，一个涉及波，但本质上都是描述系统动态的核心。

But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So for example, in classical mechanics, there's this single nervous theorem. Every time there's a symmetry in a physical system, there was a conservation law. So the laws of physics are translation invariant. Like if I move tens of the left, I experience the same laws of physics as I was here. And that corresponds to conservation momentum. If I turn around by some angle, again, I experience the same laws of physics, this corresponds to the conservation of angular momentum. If I wait for 10 minutes, I still have the same laws of physics. So there's time transition invariance, this corresponds to the law of conservation energy. So there's this fundamental connection between symmetry and conservation. And that's also true in quantum mechanics, even though the equations are completely different.

通过这个中心观念，你可以开始将许多直觉和事实从经典力学转移到量子力学中。例如，在经典力学中，有一个重要的定理叫做诺特定理。它告诉我们，每当一个物理系统中存在对称性时，就会有一个相应的守恒定律。比如，物理定律是平移不变的，即使我向左移动十米，我体验到的物理定律与之前相同，这对应于动量守恒。如果我旋转一个角度，我依然体验到相同的物理定律，这对应于角动量守恒。如果我等待十分钟，我依旧处在相同的物理定律之中，这是时间平移不变性，对应于能量守恒。因此，对称性和守恒之间存在着基本的联系。即使在量子力学中，尽管方程式完全不同，这一点仍然成立。

But because they're both coming from the Hamiltonian, Hamiltonian controls everything. Every time the Hamiltonian is a symmetry, the equations will have a conservation law. So it's, it's, it's, it's, it's, once you have the right language, it actually makes them a lot cleaner. One of the problems is why we can't unify quantum mechanics and general relativity yet. We haven't figured out what the fundamental objects are. Like, for example, we have to give up the notion of space and time being these almost cleaning type spaces. And it has to be, you know, and, you know, we kind of know that at very tiny scales, there's going to be quantum fluctuations, there's a space, space time foam. And trying to use Cartesian coordinates XYZ is going to be, it's a non-starter. But we don't know how to, what to replace it with. We don't actually have the mathematical concepts. The analog of Hamiltonian that sort of organized everything.

但是因为它们都源于哈密顿量，哈密顿量支配一切。每当哈密顿量具有对称性时，方程就会有一个守恒定律。所以，一旦你掌握了合适的语言，它们的理解就会清晰许多。一个问题在于，为什么我们还不能统一量子力学和广义相对论。我们尚未弄清楚基本的物理对象是什么。例如，我们必须放弃空间和时间是固定结构的观念。我们知道，在非常微小的尺度上，会存在量子涨落，有一种"时空泡沫"的现象。而试图使用笛卡尔的XYZ坐标系是行不通的。但我们还不知道用什么来替代它。我们实际上还没有类似于哈密顿量这样组织一切的数学概念。

Does your gut say that there is a theory of everything? So this is even possible to unify, to find this language that unifies general relativity and quantum mechanics. I believe so. I mean, the history of physics has been out of unification, much like mathematics over the years. You know, electricity and magnetism was separate theories and then backs will unify them. You know, Newton unified the motions of heavens for the motions of objects on the earth and so forth. So it should happen. It's just that the, again, to go back to this model of the observations and theory. Part of our problem is that physics is a victim of its own success. That of two big theories of physics, general relativity and quantum mechanics are so good now.

你是否直觉上认为存在一个万物理论？也就是说，是否有可能找到一种语言，将广义相对论和量子力学统一起来？我相信是有可能的。因为物理学的发展历程就是一个不断统一的过程，就像数学一样。过去，电学和磁学是分开的理论，后来人们将它们统一了起来。牛顿也实现了把天体运动和地球上物体的运动统一。在这样的背景下，统一理论是可以实现的。问题在于，物理学正因其自身的成功而面临挑战。广义相对论和量子力学这两个重要的物理理论，如今已经非常成熟。

So together, they cover 99.9% of sort of all the observations we can make. And you have to like either go to extremely insane particle celebrations or the early universe or things that are really hard to measure in order to get any deviation from either of these two theories to the point where you can figure out how to combine them together. But I have faith that we, you know, we've been doing this for centuries. We've made progress before. There's no reason why we should stop. Do you think you will be a mathematician that develops a theory of everything? What often happens is that when the physicists need some theory of mathematics, there's often some precursor that the mathematicians worked out earlier.

总之，它们几乎涵盖了我们可以进行的所有观测的99.9%。只有在极其极端的粒子碰撞实验中，或者研究早期宇宙或其他难以测量的事物时，才能发现这两种理论的偏差，以至于我们可以弄清楚如何将它们结合在一起。不过，我相信我们一直在努力探索这个领域。几个世纪以来，我们不断取得进展，没有理由停止。你认为你会成为那个发展出“万物理论”的数学家吗？通常，当物理学家需要某种数学理论时，往往是数学家之前已经做出了一些相关的前期工作。

So when Einstein started realizing that space was curved, he went to some mathematician and asked, yeah, is there some theory of curved space that the mathematicians already came up with that could be useful? And he's like, yeah, there's a, I think, a, we, we might have came up with something. And so, yeah, we might have developed a remaining geometry, which is precisely, you know, a theory of spaces that are curved in various general ways, which turn out to be almost exactly what was needed. And by Einstein's theory, there's a conductive to witness unreasonable effectiveness on mathematics. I think the theories that work well, they explain the universe tend to also involve the same mathematical objects that work well to solve the faculty problems.

当爱因斯坦开始意识到空间是弯曲的时候，他去找了一位数学家咨询，问道：“数学家们是否已经提出了关于弯曲空间的理论，可以用来帮助我呢？”那位数学家回答说：“是的，我想我们确实想出了这样一个理论。”这个理论被称为黎曼几何，它正是描述各种弯曲空间的理论，几乎完全符合爱因斯坦的需要。爱因斯坦的理论表明，在数学上有着不合理有效性的现象，即那些成功解释宇宙的理论往往涉及到能够解决抽象数学问题的相同数学对象。

Ultimately, there's just both ways of organizing data in useful ways. It just feels like you might need to go some weird land that's very hard to turn to it. Like, you have like string theory. Yeah, that was that was a leading candidate for many decades. I think it's slowly pulling out of fashion because it's not matching experiment. So one of the big challenges, of course, like you said, is experiment is very tough. Yes, because of how effective both theories are. But the other is like, just, you know, you're talking about you're not just deviating from space time. You're going into like some crazy number of dimensions.

最终，有两种方法可以以有用的方式组织数据。这感觉就像是你可能需要进入一个很难理解的奇异领域。就像弦理论一样，几个世纪以来，它曾是一个热门候选者。我觉得它正在逐渐失去青睐，因为它与实验结果不相符。当然，如你所说，其中一个巨大挑战是实验非常困难，这是因为两种理论都非常有效。但另一个挑战是，你不仅要偏离时空的概念，而是要进入一些难以想象的多维空间。

You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started. Yes, that like you mentioned. Yeah, yeah, yeah, yeah. We're just, it's very hard to use our limited, a descendants of cognition to intuit what that reality really is like. This is why analogies are so important. I mean, so yeah, the round earth is not intuitive because we're stuck on it. But, you know, but round objects in general, we have pretty good intuition. And we've introduced about light works and so forth. And it's actually a good exercise to work out how eclipses and phases of the sun and the moon and so forth. Can we really easily explain by round earth and round moon, you know, and models. And you can just take, you know, a basketball one, a golf ball, and a light source and actually do these things yourself.

你正在做各种奇怪的事情，对我们来说，我们已经离那个"平地"的起点很远了。就像你提到的，是的，是的，是的，是的。我们很难用有限的认知去直觉地理解那种现实。这就是为什么类比如此重要。比如说，圆形的地球并不直观，因为我们被困在地球表面。不过，总体来说，我们对圆形物体还是有很好的直觉理解。我们逐步了解了光的运作方式等。实际上，研究日食和月相是由圆形的地球和月亮解释的一个很好的练习。你可以用一个篮球、高尔夫球和一个光源自己实践这些现象。

So the intuition is there. But you have to transfer it. That is a big leap into lecture for us to go from flat to round earth. Because you know, our life is mostly lived in flat land. Yeah, to load that information. And we're all like to take it for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing. But, you know, we're in a round rock. Yeah, like through space. Yeah, yeah. That's a big leap. And you have to take a chain of those leaps the more and more and more we progress. Right.

所以直觉在那里。但是你必须转化这种直觉。对于我们来说，从"地球是平的"到"地球是圆的"的转变是一个巨大的飞跃。因为我们的生活大多数是在平坦的地面上进行的。是的，要掌握这个信息。我们往往对许多事情都想当然，因为科学已经为这些事情建立了很多证据。但是，你知道，我们其实是生活在一个穿越太空的圆形岩石上。是的，是的，这就是一个巨大的飞跃。而且你必须不断地进行这样的一系列飞跃，随着我们的进步越来越多。对吧。

Yeah. So modern science is maybe again a victim with its own success is that in order to be more accurate, it has to move further and further away from your initial intuition. And so for someone who hasn't gone through the whole process of science education, it looks more more suspicious because of that. So, you know, we need more grounding. I mean, I think, I mean, you know, there are scientists who do excellent outreach. But there's this, there's lots of science things that you can do at home.

“是的，所以现代科学可能再次因为其自身的成功而成为受害者。为了变得更加精确，它不得不越来越远离人们最初的直觉。因此，对那些没有经历过完整科学教育过程的人来说，它因此显得更加可疑。所以，我们需要更多的基础支持。我认为，有很多科学家在进行优秀的科普工作。同时，还有许多科学实验可以在家里进行。”

I have this lots of YouTube videos. I did a YouTube video recently of Grant Sanderson, we talked about earlier, that, you know, how the ancient Greeks were able to measure things like the distance of the moon, distance of the earth. And, you know, using techniques that you could also replicate yourself. It doesn't all have to be like fancy space telescopes and very intimidating mathematics. Yeah, that's, I highly recommend that. I believe you have a lecture and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time. Shroud and in mystery.

我有很多YouTube视频。最近我做了一个与Grant Sanderson有关的YouTube视频，我们之前讨论过，内容涉及古希腊人是如何测量像月球和地球距离这样的事物的。你知道，他们用的一些技巧其实你也可以自己尝试，不一定要用那些高级的太空望远镜或复杂难懂的数学。我强烈推荐这个视频。我相信你有一场讲座，并且还与Grant一起制作了一个出色的视频。试着将自己放在那个时代的人的思维中，是一种很美妙的体验，充满了神秘感。

You know, you're like on this planet, you don't know the shape of it, the size of it. You see some stars, you see some, you see some things and you try to like localize yourself in this world. Yeah, yeah. And try to make some kind of general statements about distance to places. Change of perspective is really important. Say, travel borders the mind. This is intellectual travel. You know, put yourself in the mind of the ancient Greeks or some other persons of other time period. Make hypotheses, spherical cows, whatever, you know, speculate.

你知道，当你站在这个星球上时，你并不了解它的形状和大小。你看到一些星星，一些东西，然后努力去在这个世界上找到自己的位置。是的，是的。同时，你试图对到各个地方的距离做出一些概括。有时改变视角非常重要。比如，人们常说旅行拓宽思维。这就像是一种智慧之旅。你可以试着把自己置于古希腊人或其他时代的人的思维中，进行假设、幻想，无论是球形奶牛还是其他的推测。

And, you know, this is what mathematicians do and some other sort of artists do, actually. It's just incredible that given the extreme constraints, you could still say very powerful things. That's why it's inspiring. Looking back in history, how much can be figured out? We don't have much. I think you're how stuff works. If you propose axioms, then the mathematics that you follow, those axioms do it to their conclusions. And sometimes you can get quite a lot, quite a long way from, you know, initial hypotheses.

你知道，这正是数学家和其他一些艺术家们所做的事情。令人难以置信的是，即使在极其有限的条件下，人们依然能表达出非常有力的观点。这也是为什么它们如此激励人心。回顾历史，能被发现的东西有多少呢？其实并不多。我认为这就是事物运作的方式。如果你提出了一些公理，那么基于这些公理的数学推导就会顺理成章地得出结论。有时候，你可以从最初的假设出发，走得很远很远。

If you're staying in the land of the weird, you mentioned general relativity. You've contributed to the mathematical understanding of Einstein's field equations. Can you explain this work? And from a sort of mathematical standpoint, what aspects of general relativity are intriguing to you, challenging to you? I have worked on some equations. There's something called the wave maps equation, all of the sigma-field model, which is not quite the equation of space-time gravity itself, but of certain fields that might exist on top of space-time.

如果你在这个奇异的领域停留过，你提到过广义相对论。你对爱因斯坦场方程的数学理解作出了贡献。你能解释一下这个工作吗？从数学角度来看，广义相对论的哪些方面对你来说是非常有趣或者具有挑战性的？我曾研究过一些方程。有一种叫做波映射方程的东西，还有σ场模型，它并不完全是描述时空引力的方程，而是描述可能存在于时空之上的某些场。

So, Einstein's equations of relativity just describe space and time itself. But then there's other fields that live on top of that. There's the electromagnetic field, there's things like Yang-Mills fields. And there's this whole hierarchy of different equations, of which Einstein's considered one of the most nonlinear and difficult. But relatively low on a hierarchy was this thing called the wave maps equation. So it's a wave, which at any given point is fixed to be like on a sphere.

爱因斯坦的相对论方程主要描述了时空本身。但在此基础上，还有其他的场存在，比如电磁场和像杨-米尔斯场这样的东西。整个方程体系中，爱因斯坦方程被认为是最非线性和最复杂的之一。但是，在这个层次结构中，相对简单的是一种叫做波映射方程的东西。这个方程描述的是一个波动，它在任何给定点上都被固定在一个类似于球体的状态。

So I can think of a bunch of arrows in space and time, and yeah, it's pointing in different directions. But they propagate like waves. If you wiggle an arrow, it will propagate and make all the arrows move kind of like a sheep's of wheat in a wheat field. And I was interested in the global or global out of problem again for this question. Like, is it possible for all the energy here? to collect at a point? So equation, I considered what's actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same. And I was able barely to show that you couldn't actually force a scenario where all the energy concentrated at one point. But at the end you had to dismiss a little bit and moment it was a little bit, it would stay regular. Yeah, this was back in 2000. That was part of why I got into the nervous talks afterwards.

所以我可以想象空间和时间中有一堆箭头，它们指向不同的方向，但是传播起来像波浪一样。如果你摆动一个箭头，它会传播出去，让所有的箭头都像麦田中的麦穗一样晃动。我对这个问题的全球性问题再次感兴趣：所有的能量能否在某一点汇聚？于是我考虑了一个被称为临界方程的方程，在这种情况下，各种尺度下的行为大致相同。我勉强能够证明你无法强迫所有能量集中在一个点的情形。但是最终你必须稍微进行一些调整，一旦这种情况发生，系统会保持稳定。那是在2000年左右，也是这之后我参加紧张交流讨论的部分原因。

Actually, yeah, so I developed some techniques to solve that problem. So part of it was, this problem is really nonlinear because of the curvature of the sphere. There was a certain nonlinear effect, which was non-perturbative. It was when you sort of looked at it normally, it looked larger than the linear effects of the wave equation. And so it was hard to keep things under control, even when the energy was small. But I developed what's called a gauge transformation. So the equation is kind of like an evolution of hives of wheat and they're all bending back and forth. And so there's a lot of motion. But if you imagine like stabilizing the flow by attaching little cameras at different points in space, which I'm trying to move in a way that captures most of the motion. And under this sort of stabilized flow, the flow becomes a lot more linear.

其实，是的，我确实开发了一些技术来解决那个问题。问题的一部分在于，由于球体的曲率，这个问题真的很非线性。存在一种特定的非线性效应，这是不可微扰的。当你通常观察这个问题时，它看起来比波动方程的线性效应要大。所以，即使能量很小，也很难控制住。但我开发了一种叫做规范变换的方法。这个方程有点像小麦在风中来回弯曲的变化过程，其中有很多运动。但你可以想象通过在空间的不同点上安装小相机来稳定流动，这些相机以某种方式移动，以捕捉绝大部分运动。在这种稳定化的流动下，流动变得更线性。

I discovered a way to transform the equation to reduce the amount of nonlinear effects. And then I was able to solve the equation. I found this transformation while visiting my art in Australia. And I was trying to understand the dynamics of all these fields. And I couldn't do a pen and paper. And I had not decided computers to do any computer simulations. So I ended up closing my eyes, being on the floor. I just imagined myself to actually be the specter field and rolling around to try to see how to change coordinates in such a way that somehow things in order of actions would behave in a reasonable linear fashion. And my aunt walked in and was doing that. And she was asking, what am I doing doing this? It's complicated. Yeah.

我发现了一种方法，可以通过转换方程，减少其中的非线性效应。这样，我就能够解出这个方程。这个发现是在我拜访身处澳大利亚的姑姑时产生的。当时，我正试图理解这些领域的动态，但用纸笔画不出结果，也没有决定使用计算机进行模拟。于是，我闭上眼睛，躺在地上，想象自己就像是一个光谱场，在地上翻滚，试图找到一种方法来改变坐标，使得事务的变化能以合理的线性方式表现出来。就在这时，我的姑姑进来了，看见我这样躺着，就问我到底在做什么，这实在是有点复杂。

And okay, fine. You're a young man. I don't ask questions. I have to ask about the, you know, how do you approach solving difficult problems? If it's possible to go inside your mind when you're thinking, are you visualizing in your mind the mathematical object symbols? Maybe what are you visualizing in your mind usually when you're thinking? A lot of pen and paper. One thing you pick up as a mathematician is sort of a collage cheating strategically. So the beauty of mathematics is that you get to change the world, change the problem, change the rules as you wish. You don't get to do this or any other field.

好的，没问题。你是个年轻人。我不问太多问题。但我必须问一下，你是如何解决困难问题的？如果可以进入你的思维，当你在思考时，你是否在脑海中想象数学对象或符号？通常你在思考时脑海中都在想些什么？用了很多笔和纸。作为数学家，你学会了一种策略性的取巧方式。数学的美妙之处在于你可以随意改变世界、改变问题、改变规则。这是在其他领域无法做到的。

Like, you know, if you're an engineer and someone says, put a bridge over this river, you can say, I want to build this up here over here instead or I want to put out a paper instead of steel. But imagine you can do whatever you want. It's like trying to solve a computer game where you can get this unlimited cheat codes available. And so, you know, you can set this, so there's a dimension that's large. I've set it to one. I'd solve the one-dimensional problem first. Or there's a main term and an error term. I'm going to make a spherical curl assumption. I'll assume the error term is zero.

如果你是一名工程师，有人让你在这条河上建一座桥，你可以说，我想在这里建一个高架桥，或者说，我想用纸而不是钢来建。试想一下，你可以随心所欲地去做，就像在玩一个电脑游戏，可以使用无限的作弊码一样。比如说，你可以设定一个很大的维度，而我把它设定为一。在这种情况下，我会先解决一维的问题。或者说，一个是主项，一个是误差项，我会假设为球形卷曲假设，并假设误差项为零。

And so, the way you solve these problems is not in sort of this ironman mode where you make things maximally difficult. But actually, the way you should approach any reasonable math problem is that you, if there are 10 things that are making it like difficult, find a version of the problem that turns off and then the difficulty is only keeps one of them. And so that. And then that just, so you install nine sheets. Okay, so 10 sheets then the game is trivial. But you install nine sheets. You solve one problem that that teaches you how to do that to get difficulty. And then you turn that one off and you turn someone else something else on and then you saw that one.

解决这些问题的方法不应该是像“钢铁侠”那样把事情搞得极其困难。实际上，处理数学问题的合理方法是：如果有10个因素让问题变得更难，试着找一个版本，把其中9个因素去掉，只保留一个困难的因素。这样做的话，在保留一个困难因素的时候，你就像给问题装上9层保护膜。假设有10层保护膜，问题就变得简单了。你在装上9层保护膜的情况下解决一个问题，这会教会你如何应对该困难因素。然后，你可以关闭这个因素，打开另一个困难因素，然后解决新的问题。

And after you know how to solve the 10 problems, 10 difficulty separately, then you have to start merging them if you had a time. I, I was a kid. I watched a lot of these Hong Kong action movies. This is from a culture. And one thing is that every time it's the fight scene, you know, something like that, the hero gets swarmed by 100 bad guy goons or whatever. But it'll always be choreographed so that you'd always be only fighting one person at a time and then you would defeat that person and move on. And because of that, they could defeat all of them. But whereas if they had fought a bit more intelligently and just swarmed the guy once, it would make for much much worse. I'm cinema, but they would win.

在你学会如何分别解决这10个问题，10种难度之后，如果有时间，你就需要开始把它们合并。我小时候看了很多香港动作电影，这是文化的一部分。有一个场景是，每次打斗时，英雄会被100个反派包围，但这些场景总是被设计成他一次只和一个人打，然后打败那个人，再继续打下一个。正因为这样，英雄才能打败所有人。但如果那些反派更聪明一点，一起包围攻击他，虽然这样电影看起来会差得多，但他们确实会赢。

Are you usually pen and paper? Are you working with computer and late tech? Mostly pen and paper actually. So in my office, I have forward giant blackboards. And sometimes I just have to write everything I know about the problem on the full blackboards and then sit on my couch and just sort of see the whole thing. Is it all symbols like notation or is there from drawings? Oh, there's a lot of drawing and a lot of bespoke doodles that only makes sense to me. And this is a bit of a blackboard you raise. It's a very organic thing. I'm beginning to use more more computers partly because AI makes it much easier to do simple coding things.

你平时倾向于使用纸笔吗？还是习惯使用电脑和现代科技？其实我大多数情况下用的是纸笔。在我的办公室里，有面大型的黑板。有时候，我需要在整块黑板上写下我对问题的所有理解，然后坐在沙发上，整理和观察整个思路。上面是符号和公式吗，还是有画图？哦，那里有很多图和只有我能理解的原创涂鸦。黑板就像一个很有机的存在。我现在开始越来越多地使用电脑了，部分原因是人工智能让一些简单的编程变得更加容易。

If I wanted to plot a function before which is moderately complicated as some iteration or something, I'd have to remember how to set up a Python program and how does a full loop work and debug it and it would take two hours and so forth. And now I can do it in 10, 15 minutes as much. I'm using more and more computers to do simple explorations. Let's talk about AI a little bit if we could. So maybe a good entry point is just talking about computer-assisted proofs in general. Can you describe the lean formal proof programming language in how it can help as a proof assistant and maybe how you started using it and how it has helped you?

如果我以前想要绘制一个相对复杂的函数，比如某个迭代，我得记住如何设置一个Python程序，以及一个完整的循环是如何工作的，还得调试，整个过程可能需要两个小时左右。而现在，我可以在10到15分钟内完成这项工作。我越来越多地使用计算机来进行简单的探索。我们可以稍微谈谈人工智能。或许一个不错的切入点是讨论一下计算机辅助证明。你能介绍一下Lean形式化证明编程语言，它如何作为辅助进行证明，以及你是如何开始使用它的，它又如何帮助了你吗？

So lean is a computer language much like sort of standard languages like Python and C and so forth. Except in most languages the focus is on using executable code. Lines of code do things. They flip bits or they make a real one move or they deliver you text on it or something. So lean is a language that can also do that. It can also be run as a standard traditional language but it can also produce certificates. So a software like Python might do a computation and give you the answer is seven. Okay, that it does the sum of three plus four is equal to seven but lean can produce not just the answer but a proof that how it got the answer of seven is three plus four and all the steps involved in.

Lean是一种计算机语言，类似于标准语言如Python和C等。然而，大多数语言的重点在于执行代码，代码行用于执行操作：可能是翻转位、让一台真实的机器运作、或者输出文本等。Lean也是一种可以执行这些操作的语言，它可以像传统语言那样运行，但它还有一个特殊的功能，那就是可以生成证明。比如，像Python这样的软件可以进行计算并给出结果，比如说3加4等于7。但是，Lean不仅能给出答案7，还能提供一个证明，说明答案7是如何得出的，以及涉及的所有步骤。

So it creates these more complicated objects not just statements but statements were proofs attached to them and every line of code is just a way of piecing together previous statements to create new ones. So the idea is not new. These things are called proof assistants and so they provide languages for which you can create quite complicated, intricate mathematical proofs and they produce these certificates that give it 100% guarantee that your arguments are correct. If you trust the compiler of the lean but they made the compiler really small and you can have several different competitors available for the same level.

因此，它不仅仅创建简单的语句，而是创建附有证明的更复杂的对象。每一行代码都是将先前的语句拼接在一起以生成新语句的方法。这个想法并不新鲜，这些工具被称为“证明助手”。它们提供了一种语言，用于创建相当复杂且精密的数学证明，并生成可以100%保证您的论点正确的证明书。如果你信任Lean的编译器，实际上他们将编译器设计得非常小，从而可以为同一层级提供几个不同的编译器。

Can you give people some intuition about the difference between writing on pen and paper versus using lean programming language? How hard is it to formalize statement? So lean a lot of mathematicians will invoke in the design of lean. So it's designed so that individual lines of code resemble individual lines of mathematical argument. You might want to introduce a variable, you want to improve our contradiction. There are various standard things that you can do and it's written so ideally it should be like a one to one correspondence.

能不能帮人们直观地理解一下，在纸上书写和使用Lean编程语言的区别是什么？将一个数学陈述形式化到底有多难？Lean的设计深受许多数学家的影响，因此它的设计目的是让每一行代码都能类似于数学推理中的每一个步骤。比如，你可能需要引入一个变量，或者需要通过反证法来证明某个命题。在Lean中有各种标准的操作，这些设计旨在实现代码与数学推理步骤之间的一一对应。

In fact it isn't because lean is like explaining a proof to an extremely pedantic colleague who will point out, did you really mean this? Like what happens if this is zero? How do you justify this? So lean has a lot of automation in it to try to be less annoying. So for example every mathematical object has to come with a type. If I talk about x, is x a rule number or a natural number or a function or something? If you write things informally it's something in terms of context. You say clearly x is equal to let x be the sum of y and z and y and z were already rule numbers. x should also be a rule number.

事实上，这并不是因为 Lean 就像是向一个非常挑剔的同事解释一个数学证明，他会指出：“你的意思真的是这样吗？”比如“如果这个数是零会怎么样？你怎么证明这一点？”因此，Lean 内置了很多自动化功能来减少这种麻烦。例如，每一个数学对象都必须有一个类型。当我提到 x 时，x 是实数、自然数、函数还是其他东西？如果你非正式地写出这些内容，通常可以通过上下文来理解。你可能会说：“显然，设 x 是 y 和 z 的和，而 y 和 z 已经是实数了，所以 x 应该也是实数。”

So lean can do a lot of that. But every so often it says wait a minute can you tell me more about what this object is? What type of object is you have to think more at a philosophical level, not just sort of computations you're doing but sort of what each object actually is in some sense. Is it using something like llm's to do the type inference or like you match with a real though? It's using much more traditional, good or fashion AI. You can represent all these things as trees and there's always algorithm to match one tree to another tree. So it's actually doable to figure out if something is a real number or a natural number. Every object sort of comes with a history of what it came from and you can kind of trace. Oh I see. Yeah. So it's designed for reliability.

所以 Lean 可以做很多这样的事情。但每隔一段时间，它会说：“等一下，你能多告诉我一些关于这个对象的信息吗？” 对象的类型需要你从哲学层面进行思考，而不仅仅是进行一些计算，而是以某种意义上理解每个对象到底是什么。它是通过使用类似于大型语言模型（LLM）的东西来进行类型推断，还是通过更传统、更靠谱的人工智能呢？其实，它使用的是更传统的人工智能方法。你可以把所有这些东西表示为树结构，然后总有算法可以将一棵树与另一棵树进行匹配。所以要弄清楚某物是实数还是自然数是可行的。每个对象都有自己的来源历史，你可以追溯其来源。哦，我明白了。是为了可靠性而设计的。

So modern AI's are not used in this is a district technology. People are beginning to use AI's on top of lean. So when a mathematician tries to program a proof in lean, often there's a step. Okay. Now I want to use the fundamental thing with calculus to do the next step. So the lean developers have built this massive project called methylib collection of tens of thousands of useful facts about methodical objects. And somewhere in there is the fundamental calculus but you need to find it. So a lot of the bottleneck now is actually lemma search. There's a tool that you know is in there somewhere and you need to find it.

因此，现代AI并未被用于这项特定技术。人们开始在Lean上使用AI。当数学家尝试在Lean中编写一个证明时，通常会有一个步骤，比如说，“现在我想使用微积分中的基本定理来进行下一步。” 为此，Lean的开发者们构建了一个庞大的项目，称为methylib，其中包含数万条关于数学对象的有用信息。在这些信息中，包含了你所需要的基本微积分内容，但你需要去找到它。因此，现在的一个主要瓶颈其实是引理搜索。工具就在某个地方，但你需要找到它。

And so there are various search engines specialized for methylib that you can do. But there's now these large language models that you can say, I need the fundamental thing with calculus at this point. And I said, okay, for example, when I code I have get up co-pilot installed as a plug-in to my IDE and it scans my text and it sees what I need. I'm not even typing. Now I need to use the fundamental thing with calculus. Okay. And then it might suggest, okay, try this and like maybe 25% of the time it works exactly and then another 10-15% of the time it doesn't quite work but it's close enough that I can say, oh, I've just changed it here and here. It will work. And then like half the time it gives me complete rubbish.

因此，有各种专门用于搜索methylib的搜索引擎。不过现在有了大型语言模型，您可以对它们说，我目前需要有关微积分的基础知识。我说，比如当我编程时，我在我的IDE中安装了GitHub Copilot作为插件，它会扫描我的文本并识别我的需求，我甚至不需要输入。这时我需要用到微积分的基础知识。好，它可能会建议我，试试这个，大约25%的时间会完全工作的很好，还有10-15%的时间可能不完全正确，但已经足够接近，让我可以稍作调整，就能很好地工作。然而，还有大约一半的时间，它给的建议完全没有用。

So but people are beginning to use AI's a little bit on top. Most of them are level of basically fancy autocomplete that you can type half of one line of a proof and it will find you'll tell you. But a fancy, especially fancy with the sort of capital letter F is remove some of the friction. Mathematician might feel when they move from patterned paper to formalizing. Yes. Yeah. So right now I estimate that the effort, time and effort taken to formalize a proof is about 10 times the amount taken to write it out. So it's doable but you don't, it's annoying. But doesn't it like kill the whole vibe of being a mathematician?

人们开始在一些地方使用AI了。多数情况下，AI的功能基本类似于高级的自动补全，你只需要输入证明的一半，AI就能帮助你完成剩下的部分。这个高级功能（尤其是如果我们用大写的F强调其高级性）能够减少数学家从手写纸张转向形式化时所遇到的一些摩擦。是的。目前我估计，将一个证明形式化所需的时间和精力大约是写出证明的10倍。所以说这虽然是可行的，但过程会有些烦人。不过，这样做是否会让做数学研究的乐趣消失呢？

Yeah. So I mean having a pedantic worker, right? Yeah. If that was the only aspect of it. Okay. But okay. There's something because it was actually more pleasant to do this formally. So there's a few of my formalized and there was a certain constant 12 that came out of it in the final statement. And so this 12 had be carried all through the proof. And like everything had to be checked then it goes. Although all these other numbers had be consistent with this final number 12. And then so we want a paper through the theorem with this number 12.

好的。我是说，有一个过分注重细节的员工，对吧？如果这只是其中的一个方面，那还好。但是，其中有一些事情让我觉得以正式的方式进行更愉快。我们做了一些形式化的工作，最后得出的结论里有一个常数12。这个12必须贯穿整个证明过程。所有的事情都必须检查，以确保其他所有数字与最后的数字12保持一致。然后我们用这个数字12完成了定理的论文。

And then a few weeks later, so I said, oh, we can actually improve this 12 to an 11 by we working some of these steps. And when this happens with pen and paper, every time you change your parameter, you have to check line by line that every single line of your proof still works. And there can be subtle things that you didn't quite realize. Some problems on number 12 that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place. So we had formalized the proof with this constant 12.

几周之后，我说，哦，我们其实可以通过调整一些步骤将这个12改进为11。当你用纸和笔进行这种改变时，每次修改参数后，你都需要逐行检查，确保你的证明中的每一行仍然有效。而有些微妙的问题可能在你没有注意到的地方潜伏着，比如在原来的12中不知不觉利用了某些性质。所以，证明可能会在某个细微之处出错。我们之前已经将12作为一个常数形式化地写入了证明中。

And then when this new paper came out, we said, oh, okay, so that took like three weeks to formalize and like 20 people to formalize this original proof. I said, oh, but now let's update the total 11. And what you can do with lean, you just in your headline theorem, you change your 12 to 11, you run the compiler. And like of the thousands of lines that code you have, 90 percent of them still work. And there's a couple that are line and red. Now I can't just buy these steps, but it meets the isolates which steps you need to change.

翻译成中文：然后当这篇新论文发表的时候，我们说，哦，好吧，所以花了大约三周时间和大约20个人来正式化这个原始证明。我说，哦，但现在我们来更新这个总数到11。使用Lean，你只需要在你的主要定理中把12改成11，然后运行编译器。在你写的成千上万行代码中，有90%仍然可以正常运行，只有几行变成了红色。我不能简单地推出这些步骤，但它能帮助你找出需要更改的步骤。

But you can skip over everything which works just fine. And if you program things correctly with good programming practices, most of your lines will not be read. And there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of you use smart tactics and so forth, you can localize the things you need to change to a very small period of time. So it's like within a day or two, we had updated our proof. Because this is a very quick process, you make a change. There are 10 things now that don't work for each one. You make a change. And now there's five more things that don't work. But the process converges much more smoothly than with pen and paper.

你可以跳过那些运行正常的部分。如果你的编程符合良好的编程习惯，大部分代码其实不需要经常阅读。只要你不对常量进行硬编码，而是采用一些聪明的方法，你就可以将需要更改的地方集中在很小的时间范围内进行处理。因此，就像我们在一两天内更新了我们的证明一样。因为这是一个非常快速的过程，你做出一个更改，可能有10个东西因此无法正常工作。你再做一个更改，又可能有五个东西不能工作。但这个过程比用纸笔处理要顺畅得多。

So that's for writing. Are you able to read it? Like if somebody else has a proof, they're able to like, how, what's the versus paper and yeah, so the proof is a longer, but each individual piece is easier to read. So if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of text or math, I often can't read it immediately. Because it assumes various definitions, which I had to go back and maybe on 10 pages earlier, this was defined. And the proof is scattered all over the place. And you basically are forced to read fairly sequentially. It's not like say a novel where like, you know, in a theory, you could you open up a novel halfway through and start reading. There's a lot of context.

所以这是关于写作的。那你能读懂它吗？比如说如果有人写了一个证明，他们能够......纸和证明有多长呢，区别是什么，对吧。证明通常会更长，但每个单独的部分更容易读懂。如果你拿着一篇数学论文，跳到第27页，读第六段，看到一些文字或数学表达式，我通常不能立刻读懂。因为它假设了很多定义，而这些定义可能在之前的10页上才有说明。证明的内容散布在整篇论文中，所以你基本上被迫按顺序阅读。这不像阅读小说，比如理论上你可以从小说的中间开始阅读，但你需要很多上下文。

But when I prove in lean, if you put your cursor on a line code, every single object there, you can hover over it and it would say what it is, what it came from, what was it justified, you can trace things back, much easier than so flipping through a math paper. So one thing that lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past. Traditionally, you pen a paper when you want to collaborate with another mathematician, either you do it at a blackboard where you can really interact. But if you're doing it sort of by email or something, basically yeah, you have to segment it. I'm going to finish section three, you do section four, but you can't really sort of work on the same thing, collaborative at the same time. But with lean, you can be trying to formalize some portion of the proof and say, I got stuck at line 67 here, I need to prove this thing, but it doesn't quite work.

但是在使用 Lean 进行证明时，当你将光标放在代码行上的任何一个对象上时，你可以悬停查看它是什么、从哪里来的、出于什么理由得到的结果。这种方法比翻阅数学论文要容易得多。因此，Lean 的一个显著优势是，它允许我们以一种前所未有的细致程度来协作进行证明。传统上，要和另一位数学家合作时，你可能会写信或者一起在黑板上互动交流。但如果你通过邮件等方式合作，你基本上需要将任务分段，比如你完成第三部分，我负责第四部分，但无法同时在同一个部分即时协作。然而，使用 Lean 时，你可以尝试形式化某个证明的一部分，并在遇到问题时，比如在第67行卡住时，指出需要证明的内容，但是工作不通。

Here's the three lines of code I can trouble with. But because all the context is there, someone else can say, oh, okay, I recognize what you need to do, you need to apply this trick or this tool. And you can do extremely atomic level conversations. So because of lean, I can collaborate with dozens of people across the world, most of them I don't have never met in person. And I may not know actually even whether they're how reliable they are in their in the proofs they can make, but lean gives me a certificate of trust. So I can do trust this mathematics.

这是我遇到麻烦的三行代码。但因为上下文都在那里，其他人可以说，哦，好吧，我知道你需要做什么了，你需要应用这个技巧或这个工具。你可以进行非常细微层次的对话。因此，由于 Lean 的存在，我可以和全世界几十个人合作，其中大多数我从未亲自见过。我可能甚至不知道他们在证明方面有多可靠，但 Lean 给了我一个信任的凭证，所以我可以信任这种数学。

So there's so many interesting questions. There's one you're known for being a great collaborator. So what is the right way to approach solving a difficult problem in mathematics when you're collaborating? Are you doing a divide and conquer type of thing or are you brain, are you focused in a particular part and your brainstorming? There's always a brainstorming process first. Yeah, so math research projects sort of by their nature, when you start, you don't really know how to do the problem. It's not like an engineering project where some other theory has been established for decades and its implementation is the main difficulty. You have to figure out even what is the right path.

有很多有趣的问题。其中一个问题是关于你被认为是一位杰出的合作者。那么，在进行合作解数学难题时，正确的途径是什么？你们是采用分而治之的方法，还是专注于某个部分进行头脑风暴？一般来说，总是先要进行头脑风暴。是的，数学研究项目本质上，在开始时你其实并不知道如何解决这个问题。它不像工程项目，其它理论可能已经被确立了几十年，主要困难在于实施。在数学研究中，你必须先弄清楚什么才是正确的路径。

So this is what I said about cheating first, you know, it's like to go back to the bridge building analogy. So first assume you have an infinite budget and like unlimited amounts of workforce and so forth. Now can you build this bridge? Okay, now have an infinite budget, but only finite workforce. Now can you do that and so forth. So of course, no engineer can actually do this because they have fixed requirements. Yes, there's this sort of jam sessions at the beginning where you try all kinds of crazy things and you make all these assumptions that aren't realistic, but you plan to fix later. And you try to see if there's even some skeleton, I'm going to push them might work.

所以这就是我关于作弊的比喻，就像造桥一样。首先，假设你有无限的预算和无限的劳动力。那么你能建这座桥吗？然后，假设你仍有无限的预算，但劳动力有限，这样你能完成吗？当然，实际上没有哪个工程师能这样做，因为他们有固定的要求。在开始阶段，会有一种“头脑风暴”的过程，你会尝试各种疯狂的想法，并假设一些不太现实的情况，计划以后再调整。这是为了看看是否有一个可能成功的框架。

And then hopefully that breaks up the problem into smaller subproblems, which you don't know how to do, but then you focus on the sub ones and sometimes different collaborators are better at working on certain things. So one of my themes I'm known for is a thing with Ben Green, which is probably Green-Tao theorem. It's a statement that the primes contain arithmetic progressions of any length. So it was a modification of this theme was already. And the way we collaborated was that Ben had already proven a similar result for progressions of length three. He showed that sets like the primes contain loss and loss of progressions of length three, even an even subsets of the primes. certain subsets do. But his techniques only worked for them three progressions. They didn't work for longer progressions.

希望通过这样的方式能把问题拆分成更小的子问题，虽然你可能不知道怎么解决这些子问题，但可以专注于其中的某些部分。有时，不同的合作者在某些方面可能更擅长。我比较知名的一个主题与本·格林（Ben Green）合作有关，那就是可能大家称为格林-陶定理（Green-Tao Theorem）的东西。该定理表述为素数包含任意长度的等差数列。这种主题已经有过一些修改。我们合作的方式是，本已经证明了关于长度为三的数列的类似结果。他证明了诸如素数这样的集合中包含大量长度为三的等差数列，甚至在素数的某些子集中也是如此。但是，他的技术只能处理长度为三的数列，不适用于更长的数列。

But I had these techniques coming from a gothic theory, which is something that I had been playing with and I knew better than I'd been at the time. And so if I could justify certain randomness properties of some set relating to primes, there's a certain technical condition, which if I could have it, if Ben could supply me this fact, I could conclude the theorem. But what I asked was a really difficult question in number theory, which he said, there's no way we can prove this. So he said, can you prove your part of the theorem using a weak hypothesis that I have a chance to prove it? And he proposed something which he could prove, but it was too weak for me. I can't use this.

我有一些技巧来源于哥特理论，这是我一直在研究并且非常熟悉的内容。因此，如果我可以证明一些与素数相关的集合的随机性特征，我就能够满足某一技术条件，从而得出定理的结论。不过，我提出的问题在数论中非常困难，Ben告诉我："我们不可能证明这个。" 于是，他问我能否利用他可能有机会证明的较弱假设去完成定理的证明。他提出了一个他可以证明的假设，但对我来说太弱了，无法使用。

So there's this conversation going back and forth. So the different cheats too. Yeah, I want to cheat more than he wants to cheat less. But eventually we found a property which A he could prove in B I could use. And then we could prove that to you. And so there's a there's a there all kinds of dynamics, you know, I mean, it's every collaboration has a has a has some story. It's no two of the same. And then on the flip side of that, like you mentioned, with lean programming, now that's almost like a different story because you can do you can create, I think you've mentioned a kind of a blueprint, right, for a problem.

这段话的大意是说，两个人在合作过程中有来回讨论，各自对如何“投机取巧”有不同的看法。其中一个人希望多应用技巧，而另一个人希望少用。最终，他们找到了一种属性，其中一个人能够证明这种属性，而另一个人能够加以利用，从而一起成功解决问题。每次合作都会有自己独特的故事，没有两次是完全相同的。另一方面，就像提到精益编程那样，这又是一个不同的故事。通过这种方法，你可以为解决问题创建一个“蓝图”。

And then you can really do a divide and conquer with lean, where you're working on separate parts. Right. And they're using the computer system proof checker essentially to make sure that everything is correct along the way. So it makes everything compatible and yeah, and trustable. Yeah. So currently, only a few mathematical projects can be cut up in this way. At the current state of the art, most of the lean activity is on formalizing proofs that have already been proven by humans. And math paper basically is a boop a blueprint in a sense. It is taking a difficult statement like big theorem and breaking it up into me are 100 little numbers.

然后，你可以用精益的方法进行分而治之，也就是处理不同的部分。对，就是这样。他们其实在使用计算机系统的证明检查器来确保整个过程中一切都是正确的。所以这使得所有内容都兼容，而且值得信赖。目前，只有少数数学项目能够以这种方式拆分。根据当前的技术水平，大多数精益活动集中在形式化那些已经由人类证明过的证明。简单来说，数学论文就像是一种蓝图。它把一个困难的命题，比如一个重要的定理，分解成许多小的部分来处理。

But often not all written with enough detail that each one can be sort of directly formalized. A blueprint is like a really pedantically written version of a paper where every step is explained as much detail as as as possible. And to try to make each step kind of self-contained. And depending on only a very specific number of previous statements, I mean, proven so that each node of this blueprint graph that gets generated can be tackled independently of all the others. And you don't even need to know how the whole thing works. So it's like a modern supply chain. You know, like if you want to create an iPhone or some other complicated object, no one person can build a single object.

然而，通常这些内容没有详细到能够直接形式化的程度。蓝图就像是一篇写得非常细致的论文，其中每一个步骤都被详细解释，并尽量做到每一步都是自成一体的，仅依赖于之前少量的陈述。这意味着生成的蓝图图谱中的每个节点都可以独立处理，而不需要了解整个系统的运作方式。这就像是现代的供应链一样，比如说如果你想制造一部iPhone或其他复杂的物品，没有一个人能独自完成它。

But you can have a specialist who just if they're given some widgets from a similar company, they can combine them together to form a slightly bigger widget. I think there's a really exciting possibility because you can have if you can find problems that could be broken down this way, then you can have thousands of contributors, right? Yes, yes, yes, distributed. So I told you before about the split between theoretical and experimental mathematics. And right now, most mathematics is theoretical and when you type it is experimental. I think the platform that lean and other software tools, so get hub and things like that, allow experimental mathematics to be to scale up to a much greater degree than we can do now.

但你可以找一个专家，如果他们拿到一些来自类似公司的小部件，他们可以把这些小部件组合在一起，形成一个稍微大一点的部件。我认为这是一个非常令人兴奋的可能性，因为如果你能找到可以这样拆解的问题，那么你就可以有成千上万的贡献者，是的，是的，是分布式的。我之前已经告诉过你理论数学和实验数学之间的区别。现在，大多数数学是理论性的，而当你输入时，它就是实验性的。我认为像 Lean 这样的平台以及其他软件工具，例如 GitHub 等，使得实验数学可以规模化到比我们现在能够做到的程度更大的水平。

So right now, if you want to do any mathematical exploration of some mathematical pattern or some of you need some code, do write out the pattern. And I mean, sometimes there are some computer algebra packages that help. But often it's just one mathematician coding lots and lots of Python or whatever. And because coding is such an error for an activity, it's not practical to allow other people to collaborate with you on writing modules for your code, because if one of the modules has a bug in it, the whole thing is unreliable. So you get these Spock spaghetti code written by non-operational programmers with mathematicians, and they're clunky and slow.

现在，如果你想研究一些数学模式或者需要一些代码来写出这个模式，你可以考虑自己动手编写代码。有时候，确实有一些计算机代数软件包可以提供帮助。但往往情况是，只有一个数学家在用Python或者其他语言编写大量代码。因为编写代码本身是一个容易出错的活动，要让别人和你一起协作编写代码模块并不实际，因为如果其中一个模块有错误，整个程序就会不可靠。因此，你会看到这些由数学家编写的“意大利面”式混乱代码，它们既笨拙又运行缓慢。

And so because of that, it's hard to really mass-produce experimental results. But I think with lean, I mean, I'm already starting some projects where we are not just experimenting with data, but experimenting with proofs. So I have this project called the Equational Theory's project. Basically, we generated about 22 million little problems at abstract algebra. We should back up and tell you what the project is. Okay, so abstract algebra studies operations like multiplication and addition and the abstract properties. Okay, so multiplication, for example, is commutative. X times Y is always Y times X is for numbers. And it's also associative. X times Y times Z is the same as X times Y times Z. So these operations are based on laws that don't obey others. For example, X times X is not always equal to X. So that laws are not always true.

所以，由于这个原因，很难真正大规模地生产实验结果。但我认为通过精益方法，我实际上已经开始了一些项目，在这些项目中，我们不仅对数据进行实验，还对证明进行实验。我有一个名为“方程理论项目”的计划。基本上，我们在抽象代数领域生成了大约2200万个小问题。我们应该退一步来解释这个项目是什么。好的，抽象代数研究的是像乘法和加法这样的运算及其抽象性质。举个例子，乘法是可交换的，也就是说对于数字来说，X乘以Y总是等于Y乘以X。它也是结合的，X乘以Y再乘以Z与X乘以Y乘以Z是相同的。这些运算基于一些不遵循其他规律的规则。例如，X乘以X并不总是等于X。因此，这些规律并不总是成立的。

So given any operation, it obeys some laws and not others. And so we generated about 4,000 of these possible laws of algebra that certain operations can satisfy. And our question is which laws imply which other ones. So for example, does commutativity imply associativity? And the answer is no, because it turns out you can describe an operation which obeys the commutative law, but it doesn't obey the associative law. So by producing an example, you can show that commutativity does not imply associativity. But some of the laws do imply other laws by substitution and so forth. And you can write down some algebraic proofs. So we look at all the pairs between these 4,000 laws and the sort of 22 million of these pairs. And for each pair, we ask, does this law imply this law? If so, give a proof. If not, give a count example.

任何一个运算都遵循某些法则，而不遵循另一些法则。因此，我们生成了大约4,000个这样的代数法则，某些运算可以满足这些法则。我们的问题是，哪些法则能够推出其他哪些法则。例如，交换律能否推出结合律？答案是否定的，因为你可以描述一种运算，它符合交换律但不符合结合律。通过举例，你可以证明交换律不能推出结合律。然而，有些法则可以通过代入等方式推出其他法则，并可以写成一些代数证明。因此，我们查看这4,000个法则之间的组合，大约有2,200万对。对于每一对，我们都要问，这个法则能否推出那个法则？如果可以，请给出证明；如果不可以，请举出反例。

So 22 million problems, each one of which you could give to an undergraduate algebra student. And they had a decent chance of solving the problem. Although there are a few, at least 22 million, there are like 100 or so that are really quite hard. But a lot are easy. And the project was just to work out to determine the entire graph, like which ones imply which other ones. That's an incredible project, by the way. Such a good idea, such a good test that the very thing we've been talking about at a scale that's remarkable.

所以有2200万个问题，其中每一个问题都可以交给一个本科生来解决，他们很有可能能够解决这些问题。尽管其中至少有一百个左右问题是相当困难的，但大多数都比较简单。这个项目的目的就是要弄清楚整个图（graph），也就是哪些问题可以推导出其他问题。顺便说一下，这是一个了不起的项目。如此好的主意，这么好的测试，以一种极其惊人的规模实现了我们一直在谈论的东西。

Yeah, so it would not have been feasible. I mean, the state of the art in the literature was like 15 equations and sort of how they imply that sort of the limit of what a human repentant people can do. So you need to scale that up. So you need to crowdsource, but you also need to trust all the, no one person can check 22 million of these proofs. You need to be computerized. And so it only became possible with lean. We were hoping to use a lot of AI as well. So the party is almost complete. So all these 20 million, all but two have been settled.

是的，所以这是不可行的。我是说，文献中最先进的研究大概涉及15个方程，而这基本上代表了人类所能达到的极限。因此，你需要扩大规模。你需要众包，但也必须信任，因为没有人能亲自检查2200万个这样的证明。所以你需要借助计算机。因此，这只有通过Lean才成为可能。我们也希望大量使用人工智能。现在几乎完成了所有任务，2200万中的所有内容，除了两个，都已经解决了。

Well, actually, and all those two, we have a pen and paper proof of the two. And we're formalizing. In fact, I was this morning, I was working on it, finishing it. So we're almost done on this. It's incredible. It's yeah, a fact is how many people were able to get 50, which in mathematics is considered a huge number. It's a huge number. Yeah, crazy.

好的，实际上，我们已经有了那两个问题的纸笔证明，并且正在将其格式化。事实上，今早我还在处理这个问题，快要完成了。这真的很不可思议。是的，这个事实是，有多少人能够得到50，在数学中被认为是一个非常大的数字。是的，太疯狂了。

Yeah, so we got about paper 50, all this and a big appendix of food contributor. What? Here's an interesting question. Not to maybe speak even more generally about it. When you have this pool of people, is there a way to organize the contributions by level of Vux Partisa, the people of the contributors? Okay. I'm asking you a lot of pot head questions here, but I'm imagining a bunch of humans and maybe in the future, some AI's.

好的，我们有大约50页的文件，以及一个关于食物贡献者的大附录。你提出了一个有趣的问题，也许可以更广泛地讨论一下。当你有这样一组人时，有没有办法根据贡献者的Vux Partisa水平来组织这些贡献？好吧，我知道我问了很多让人犯迷糊的问题，但我在想象一群人，也许未来还会有一些人工智能。

Yeah. Can there be like an elo rating type of situation where like a gamification of this, the beauty of these lean projects is automatically you get all this data. Yeah. So like everything's we uploaded with this guitar and GitHub tracks who contributed what? So you could generate statistics from at any, at any later point in time, you could say, oh, this person contributed this made this many lines of code or whatever. I mean, these are very crude metrics. I would, I would definitely not want this to become like part of your tenure review or something.

好的。这种情况可以像 Elo 评分那样进行游戏化吗？精益项目的美妙之处在于，你可以自动获得所有这些数据。所有东西都可以通过吉他和 GitHub 上传，它会追踪每个人的贡献情况。这样的话，你就可以在任何时候生成统计数据，比如说，这个人贡献了多少代码行数等。不过，这些都是很粗略的指标，我绝对不希望这些成为你评估任期的一部分。

But I mean, I think already in enterprise computing, right? People do use some of these metrics as part of the assessment of performance of an employee. Again, this is the direction which is a bit scary for academics to go down. We don't like metrics so much. And yet academics use metrics. They just use old ones. Number of papers.

我的意思是，我想在企业计算中，这已经很常见了，对吧？人们确实使用一些指标来评估员工的表现。同样地，这种趋势对学术界来说有点让人担忧。我们学术界并不太喜欢指标。不过，学者们也使用指标，只是用的是老旧的指标，比如论文数量。

Yeah. It's true. It's true that I mean, it feels like this is a metric while flawed is going in the more in the right direction, right? Yeah. It's interesting. At least it's a very interesting metric. Yeah. I think it's interesting to study. I mean, I think you can do studies of whether these are better predictors. There's this problem called good hard slow. If a statistic is actually used to incentivize performance, it becomes gained. And then it is no longer a useful measure.

是的，确实如此。我的意思是，虽然这个指标有缺陷，但它似乎正朝着更正确的方向发展。至少这是一个非常有趣的指标。我认为研究它很有意思。我觉得可以研究一下这些指标是否可以成为更好的预测工具。有一个问题叫做 "古德哈特定律"（Goodhart's Law），如果一个统计指标被用来激励表现，人们就会开始操纵它，然后这个指标就不再是一个有用的衡量标准了。

Oh, humans always. Yeah. Yeah. I mean, it's rational. So what we've done for this project is self-report. So there are actually standard categories from the sciences of what types of contributions people give. So there's this concept and validation and resources and coding and so forth. So we, we, there's a standard list of pro or so categories. And we just ask each contributor to this big matrix of all the, all the, all the, all the categories just to tick the boxes where they think they're contributed. And just give a rough idea, you know, like, oh, so you did some coding and, and, and you provided some compute, but you didn't do an A for pen and paper verification or whatever. And I think that that works out.

哦，人类总是这样。是的，是的。我是说，这很合乎逻辑。对于这个项目，我们所做的是自我报告。其实在科学领域有标准的类别，用来分类人们所作的贡献类型。这些类别包括概念、验证、资源、编程等等。因此，我们有一个标准的类别清单。我们让每位参与者在这个大矩阵中选择他们认为自己有贡献的类别。这样就可以大致了解，比如你做了一些编程，也提供了一些计算资源，但你没有进行纸和笔的验证之类的。我认为这样的方式效果不错。

Traditionally, mathematicians just order alphabetically by surname. So we don't have this tradition as in their sciences of, you know, lead author and second author and so forth. Like, which we're proud of, you know, we make all the authors equal status, but it doesn't quite scale to this size. So a decade ago, I was involved in these things called polymath projects. It was the crowdsourcing mathematics, but without the lean component. So it was limited by, you needed a human moderator to actually check that all the contributions coming in were actually valid. And this was a huge bottom neck, actually. But still, we had projects that were, you know, 10 authors or so, but we had decided at the time not to try to decide who did what, but to have a single pseudonym.

传统上，数学家们习惯用姓氏的字母顺序来排列作者名次。因此，我们不像其他科学领域那样有第一作者、第二作者之类的传统。这也是我们的一个特点，我们为此感到自豪，因为我们让所有的作者享有同等的地位。但是这种方式在面对大规模项目时效果并不理想。大约十年前，我参与了一些所谓的“多元数学项目”，这是一种众包数学问题的尝试，但不包括简化后的部分。因此，它受到了一些限制，比如需要人为的审核员来确认所有提交的贡献都是有效的，这实际上是个很大的瓶颈。不过，即使是这样，我们的项目通常会有大约10位作者，但我们当时决定不去区分每个人的具体贡献，而是用一个共同的化名。

So we created this fictional character called DHJ Polymath in the spirit of Bourbon by Keywarkies is the pseudonym for famous group of mathematicians in the 20th century. But, and so the paper was altered on the pseudonym. So none of us got the author credit. This actually turned out to be not so great for a couple of reasons. So one is that if you actually wanted to be considered for 10 year or whatever, you could not use this paper in your, as you submitted, as many publications because it didn't have the formal author credit. But the other thing that we've recognized a little much later is that when people referred to these projects, they naturally refer to the most famous person who was involved in the project. Oh, so this was Tim Gowas, this was Tim Gowas project. This was Tim Gowas project. And not mention the other 19 or whatever people that were involved.

我们创建了一个虚构角色，名叫DHJ Polymath，这个名字的灵感来源于Keywarkies笔名下的20世纪著名数学家群体。但是，论文是在这个笔名下发表的，所以我们没有人获得作者署名。出于几个原因，这实际上结果并不太好。首先，如果你希望获得终身教职或其他职位，你不能将这篇论文作为你提交的多项出版物之一，因为它没有正式的作者署名。后来我们还发现，当人们提到这些项目时，他们自然会提到项目中最著名的人。哦，这是Tim Gowers的项目，这是Tim Gowers的项目，却没有提到其他参与的19个人或更多人。

So we're trying something different this time around where we have everyone's an author. But we will have an appendix with this matrix. And we'll see how that works. I mean, so both projects are incredible just the fact that you're involved in such huge collaborations. But I think I saw a talk from Kevin Buzzer about the lean programming language is a few years ago and you're saying that this might be the future of mathematics. And so it's also exciting that you're embracing one of the greatest mathematicians in the world embracing this, what seems like the paving of the future of mathematics.

这次，我们尝试一些不同的做法，让人人都是作者。但我们会附上一个矩阵作为附录，我们会看看这样效果如何。我是说，这两个项目都非常了不起，仅仅是因为你们参与了如此大型的合作。然而，我想起几年前听过Kevin Buzzer关于Lean编程语言的演讲，他提到这可能是数学的未来。所以，你们正参与一位世界上最伟大的数学家所认同的、似乎是数学未来发展方向的项目，这让人感到非常激动。

So I have to ask you here about the integration of AI into this whole process. So DeepMai's Alpha Proof was trained using reinforcement learning on both failed and successful formal lean proofs of IMO problems. So this is sort of high level high school. Oh, very high level, yes. Very high level high school level mathematics problems. What do you think about the system? And maybe what is the gap between this system that is able to prove the high school level problems versus gradual level problems?

所以我得在这里问你一下关于将人工智能融入整个过程的情况。DeepMai 的 Alpha Proof 使用强化学习训练，基于国际数学奥林匹克竞赛（IMO）问题的成功和失败的形式化 Lean 证明。这其实相当于高中很高水平的数学问题哦。是的，非常高水平的高中数学问题。你对这个系统怎么看？也许你能谈谈这个系统能够证明这些高水平高中数学问题和能解决大学研究生水平数学问题之间的差距是什么？

Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It's a commentarial explosion. So I think of large language models is that they make mistakes. And so if your proof has got 20 steps and your art line board has a 10% failure rate at each step of going in the wrong direction. It's extremely unlikely to actually reach the end. Actually, just to take a small tangent here, how hard is the problem of mapping from natural language to the formal program?

是的，随着证明步骤数量的增加，难度呈指数级上升。这就像是一种注释爆炸现象。因此，我认为大型语言模型会犯错。如果你的证明包含20个步骤，而你的人工智能模型在每个步骤出错的概率是10%，那么它实际上很难顺利到达终点。顺便说一下，从自然语言到形式化程序的映射问题到底有多难呢？

Oh, yeah, it's extremely hard actually. Natural language, you know, it's very full-tolerant. Like you can make a few minor grammatical errors and speak in the second language, you can get some idea of what you're saying. But formal language, if you get one little thing wrong, I do that the whole thing is nonsense. Even formal to formal is very hard. There are different incompatible prefaces in languages. There's lean but also cock and Isabelle and so forth. And I keep even converting from a formal language to formal language. It's an unsolved miscalusable problem. That is fascinating.

哦，是的，这其实非常难。自然语言非常具有容错性。比如你在说第二语言时，即使有一些小的语法错误，大家还是可以大致明白你的意思。但在形式语言中，如果你有一点小错误，整个东西可能就变得毫无意义了。从一种形式语言转换到另一种形式语言也很困难。不同的语言有各自不兼容的前缀，比如Lean、Coq和Isabelle等等。我一直在尝试将一种形式语言转换为另一种形式语言，这仍然是一个未解决的问题。这真的很有趣。

Okay, so but once you have an informal language, they're using their RL-trained model. So something akin to Alpha 0 that they used to go to then try to come up with proofs. They also have a model, I believe, is a separate model for geometric problems. So what impresses you about the system and what do you think is the gap? We talked earlier about things that are amazing over time become kind of normalized. So now somehow of course geometry is still available from... Right, that's true, that's true. I mean it's still beautiful. Yeah, yeah, it's a great work. It shows what's possible. The approach doesn't scale currently. Three days of Google's service, server time to do one high school math formula. This is not a scalable plus spect, especially with the exponential increase in complexity increases.

好的，所以一旦你拥有了一种非正式语言，他们就使用他们的强化学习训练模型。有点类似于他们用于 Alpha 0 的模型，然后尝试提出证明。我认为，他们也有一个针对几何问题的独立模型。那么，这个系统让你印象深刻的是什么？你认为缺陷在哪里？我们之前谈到，有些令人惊叹的事物随着时间推移变得平常。那么现在，几何学当然仍然可用……对的，那是真的，那是真的。我是说它依然很美。是的，是的，这是一个很好的工作，展示了可能实现的东西。这个方法目前并不能很好地扩展。使用三天的 Google 服务器时间来解决一个高中数学公式，这种方法不具备可扩展性，特别是随着复杂性的指数级增长。

Which mentioned that they got a silver medal performance. The equivalent of... I mean, yeah, the equivalent of... Yeah, I mean, they... So first of all, they took way more time than was allotted and they had this assistance where the humans started helped by formalizing. But also they're giving us those formats for the solution, which I guess is formally verified. So I guess that's fair. There are efforts, there will be a proposal at some point to actually have an AI method of the year where at the same time as the human contestants get the actual little bit problems, yes, we'll also be given the same problems at the same time period. And the outputs will have to be created by the same judges. Which means that we'll have to be written in natural language rather than formal language. I hope that happens. I hope that this time won't happen.

这段英文提到他们获得了银牌表现。相当于……我的意思是，他们……首先，他们花的时间比分配的多得多，并且人类在开始时通过形式化提供了一些帮助。此外，他们给我们提供了那些解决方案的格式，我猜这些方案是经过正式验证的，所以我觉得这很合理。有一些努力，未来某个时候会有一个提案，实际上设立一个年度AI方法奖。在这个奖项中，AI方法会和人类参赛者同时获得相同的问题，解答输出将由相同的评委来评判。这意味着解答需要用自然语言而不是形式化语言书写。我希望这能实现。我希望这不会发生在未来。

I hope that's fine. It won't happen this IMO. The performance is not good enough in the time period. But there are smaller competitions. The competitions where the answer is a number rather than a long form proof. And that's... AI is actually a lot better at problems where there's a specific numerical answer. Because it's easy to reinforce learning on it. You've got the right answer, you've got the wrong answer. It's a very clear signal. But a long form proof either has to be formal and then the lean can give it thumbs up, thumbs down. Or it's informal. But then you need a human to create it. And if you try to do billions of reinforcement learning, you can't hire enough humans to create those.

我希望这样可以。这次不太可能实现。表现不够好，因为时间有限。不过还有一些小型比赛。在这些比赛中，答案是一个具体的数字，而不是长篇证明。在这方面，AI其实更擅长，因为这样很容易进行强化学习。答案是对还是错，这个信号非常明确。但对于长篇证明，要么需要正式的形式，Lean可以给出通过或不通过的评价；要么是非正式的，但这就需要人类来创建。如果尝试进行数十亿次的强化学习，你无法雇佣足够的人来创建这些。

It's very hard enough for the last time criminals to do reinforcement learning on just the regular text that people get. But now if you hire people not just give thumbs up thumbs down, but actually check the output mathematically. That's too expensive. So if we explore this possible future, what is the thing that humans do that's most special in mathematics? So that you could see AI not cracking for a while. So inventing new theories. So coming up with new conjectures versus proving the conjectures, building new abstractions, new representations, maybe an AI dinosaur with seeing new connections between disparate fields. That's a good question. I think the nature of what mathematicians do over time has changed a lot.

要让那些曾经犯罪的人仅仅从普通文本中学会强化学习已经非常困难。但如果你雇人不仅仅是给内容点赞或踩，还要实际地从数学上检查输出结果，那就太昂贵了。如果我们探讨这个可能的未来，人类在数学上做什么才是最特别的呢？也许在一段时间内，AI在这方面还无法赶上人类。比如，发明新理论、提出新猜想、证明猜想、建立新的抽象概念、新的表示方法，或者像AI恐龙那样在不同领域之间看到新的联系。这是一个很好的问题。我认为随着时间的推移，数学家所做的事情的本质已经发生了很大的变化。

So a thousand years ago, mathematicians had to compute the date of Easter. And then we had complicated calculations. But it's all automated. It's been an automated centuries. We don't need that anymore. They used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from the old board to the new. So I think it's very complicated calculations. Again, we'd been automated. Even a lot of undergraduate mathematics, even before AI, like both from alpha, for example, it's not a language model. But you can solve a lot of undergraduate level f-tasks. So on the computational side, verifying routine things, like having a problem and say, here's a problem in partial differential equations.

大约一千年前，数学家们需要计算复活节的日期，那时需要进行复杂的计算。但现在这一切都已实现自动化。这种自动化已经持续了几个世纪，不再需要手动计算了。过去，人们要用球面导航和球面三角学来确定从旧大陆航行到新大陆的路线。这也是非常复杂的计算，同样已经自动化。即使在人工智能出现之前，比如像 Wolfram Alpha 这样的工具，也能解决很多大学本科水平的数学任务。在计算方面，验证一些常规的问题变得更加轻松，比如给出一个偏微分方程的问题。

Could you solve it using any of the 20 standard techniques? And they have a yes, I've tried all 20 and hear that 100 different permutations and disease map results. And that type of thing, I think, it worked very well. Type of scaling to, once you solve one problem to make the AI attack 100 adjacent problems, the things that humans do still, where the AI really struggles right now, is knowing when it's made a wrong turn. And you can say, oh, I'm going to solve this problem. I'm going to split up this woman into these two cases. I'm going to try this technique. And sometimes, if you're lucky, it's a simple problem. It's the right technique and you solve the problem.

你能用20种标准技术中的任何一种来解决这个问题吗？他们会说可以，我尝试过所有20种方法，并听说有100种不同的排列和疾病地图结果。类似这样的事情，我认为效果很好。这种方法的扩展性在于，一旦你解决了一个问题，就可以让AI处理100个相邻的问题。目前AI真正困难之处在于，它不知道什么时候做了错误的选择。而人们依然在做的事情是，比如说，我要解决这个问题，我将把情况分为两种进行尝试不同的技术。有时候，如果运气好，问题很简单，你使用了正确的技术，就能解决问题。

Sometimes it will have a problem. It would propose an approach which is just complete nonsense. But it looks like a proof. So this is one annoying thing about LLM generated mathematics. So we've had human general mathematics as a very low quality. Submissions, we don't have the formal training and so on. But if a human proof is bad, you can tell it's bad pretty quickly. It makes really basic mistakes. But the AI general proofs they can look superficially flawless. And it's partly because that's what the reinforcement learning has, like you train them to do, to make things to produce text that looks like what is correct. But for many applications, it's good enough. So it was often really subtle. And then when you spot them, they're really stupid. Like no human would have vacuumed that mistake.

有时候，它会出现问题，提出一些完全无意义的方法。但这些方法看起来像是有理有据的证明。这是让人恼火的地方之一，关于大语言模型生成的数学内容。我们都知道人类的数学普遍存在质量参差不齐的情况，一些投稿质量很低，可能是因为缺乏正规训练等等。但如果一个人类的数学证明很差，你很快就能看出问题，因为它会犯非常基础的错误。而人工智能生成的证明从表面上看可能无懈可击，这部分是因为强化学习中训练它们要生成看起来正确的文字。对于许多应用来说，这种表现已经够好了。不过，这些错误常常是非常细微的，当你最终发现它们时，会觉得非常愚蠢，像是人类绝不会犯的错误。

Yeah, it's actually really frustrating in the programming context because I program a lot. And yeah, when a human makes low quality code, there's something called code smell. Right? You can tell. You can tell immediately. Like, yeah, there's signs. But with AI generated code, it's old of us. And then you're right. Eventually you find an obvious dumb thing that just looks like good code. Yeah. So it's very tricky to and frustrating for some reason to. Yeah. So yeah. So the sense of smell. This is one thing that humans have. And there's a metaphor called mathematical smell that is not clear how to get there. You have to do pretty well. Eventually, I mean, so the way Alpha Zero and software to make progress and go and chest and so all this is in some sense they have developed a sense of smell for go and chest positions.

在编程环境中，这确实让人非常沮丧，因为我编程很多。是的，当人们写出低质量的代码时，有一种东西叫做代码异味，你能立即察觉到一些迹象。但是对于AI生成的代码来说，这种代码异味就不容易察觉。等到你发现问题时，往往是因为代码中有明显愚蠢的错误，看起来很像是没问题的代码。对这种情况非常棘手和沮丧。人类有一种天生的洞察力，可以通过“闻到”代码的异味来识别问题。但对于数学上的异味这种比喻来说，目前还不清楚如何达到这样的洞察水平。像AlphaZero这样的软件在围棋和国际象棋上取得了进展，某种程度上是因为它们在这些棋局中发展出了“嗅觉”，能够以某种直觉的方式找到解决方法。

That this position is good for white. That's good for black. They can't initiate why. But just having that sense of smell lets them strategize. So if AIs gained that ability to sort of a sense of viability of certain proof strategies, so you can say, I'm going to try to break up this problem into two small sub tasks and then you can say, oh, this looks good. The two tasks look like they're simpler tasks than your main task. And they still got a good chance of being true.

这个局面对白方有利。对于黑方来说，这是一个好局面。他们可能说不出具体原因，但凭借这种直觉，他们可以制定策略。如果人工智能也能获得这种能力，能够对某些证明策略的可行性有一种直觉，那么你可以说，我要尝试将这个问题分解为两个小的子任务，然后你可以认为，这样看起来不错。这两个子任务比主要任务简单，而且仍有很大的可能性能够实现。

So this is good to try. Or you've made the problem worse because each of the two sub problems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try. Normally, it's very easy to transform a problem into a even harder problem. Very rarely do you transform a simpler problem. So if they can pick up a sense of smell, then they could maybe start competing with human law mathematicians. So this is a hard question, but not competing, but collaborating. If, okay, hypothetical, if I gave you an oracle that was able to do some aspect of what you do and you could just collaborate with it.

所以这是值得尝试的。如果不然，你可能会把问题变得更糟，因为把原来的问题拆分成两个子问题，通常情况下每个子问题都会比原来的问题更难。通常情况下，你尝试随便找个方法解决问题时，很容易把问题变得更复杂，而很少会把它变得更简单。所以，如果他们可以拥有一种“嗅觉”的能力，那么他们可能会开始与人类数学家进行竞争。不过，这里并不是说竞争，而是合作。假设有一个能帮你完成某些工作的“神谕”，你可以与它合作，这就是一个难题。

Yeah, yeah. What would that oracle, what would you like that oracle to be able to do? Would you like it to maybe be a verifier, like check to the code smut, like your, yes, a professor, child, this is the correct, this is a good, this is a promising fruitful direction.

好的，好的。你希望那个神谕能够做些什么呢？你会希望它像一个验证者一样，比如检查代码，确认是否正确吗？就像一个教授对学生说：“这是正确的，这是好的，这是一个有前景和有成果的方向。”

Yeah, yeah, or would you like it to generate possible proofs and then you see which one is the right one? Or would you like it to maybe generate different representation, totally different ways of seeing this problem? I think all of the above. A lot of it is, we don't know how to use these tools because it's a paradigm that it's not, yeah, we have not had in the past assistance that are competent enough to understand complex instructions that can work at massive scale, but also unreliable. It's an interesting, a bit unreliable in subtle ways, was we was sufficiently good output.

是的，是的，你是想让它生成可能的证明，然后你再查看哪个是正确的？还是你希望它能生成不同的表现方式，或者用完全不同的视角来看待这个问题？我认为以上都是有可能的。很多时候，我们不知道如何使用这些工具，因为这是一种新的模式。我们过去并没有这样的助手，它们足够聪明，可以理解复杂的指令，并在大规模上工作，但同时也不够可靠。这种兼具良好输出和细微不可靠性的助手是一种有趣的现象。

It's an interesting combination. You have like graduate students who work with who kind of like this, but not as scale. And we had previous software tools that can work at scale, but very narrow. So we have to figure out how to use, I mean, so Tim Kowd, you imagine, Yakis for saw, like in 2000, he was envisioning what mathematics would look like in, like, two and a half decades.

这是一种有趣的组合。你有一些研究生，他们在和这种类似的东西一起工作，但规模不大。我们以前有一些可以大规模工作的软件工具，但它们非常局限。所以我们需要弄清楚如何使用它。你可以想象一下，Tim Kowd 在2000年时设想未来二十五年后的数学发展会是怎样的。

Yeah, he wrote in his article, like a hypothetical conversation between a mathematical assistant of the future and himself, you know, a pharmaceutical problem, and they would have a conversation that sometimes the human would propose an idea and the AI would evaluate it. Sometimes the AI would propose an idea, and sometimes the computational is required and the AI would just go and say, okay, I've checked the 100 cases needed here, or the first, you set the situation for all end up checking for end up to 100, and it looks good so far, or hang on, there's a problem that equals 46.

在他的文章中，他写了一段未来数学助手和他之间的假想对话，你知道，就是一个有关药物的难题。对话中，有时候人类会提出一个想法，然后AI会进行评估。有时候是AI提出一个想法，而当需要计算时，AI会直接说，比如说，我已经检查了这里需要的100个案例，或者你设置的条件检查到了100，现在看起来还不错，或者说等等，这里有一个问题在等于46时出现。

And so just a free form conversation where you don't know in advance where things are going to go, but just based on, I think, ideas that get proposed on both sides, calculations get proposed on both sides, I've had conversations with AI, where I say, let's, we're going to collaborate to solve this math problem. And it's a problem that I already know the solution to. So I try to prompt it. Okay, so here's the problem. I suggest using this tool, and it'll find this lovely argument using it from a different tool, which eventually goes into the weeds and say, no, no, no, if I using this, okay, and I start using this, and then it'll go back to the tool that I wanted to do before. And you have to keep railroading it onto the path you want, and I could eventually force it to give the proof I wanted. But it was like hurting cats, and the amount of personal effort I had to take to, not just to prompt it, but I also check it output, because a lot of what it looks like is going to work, I know there's a problem on 917, and basically arguing with it. It was more exhausting than doing it on the system.

这段话的大意是描述了与人工智能进行自由对话时的一种体验。这种对话没有事先安排的方向，而是基于双方提出来的想法与计算进行讨论。作者举了一个例子：他与人工智能合作解决一个数学问题，而这个问题的答案他已经知道。他尝试引导人工智能解决问题，提供了一个工具作为提示，但人工智能使用了另一个工具，并因此产生了一段复杂的讨论。作者努力让人工智能回到他希望的方法上。最终，他可以迫使人工智能提供他想要的证明过程，但这就像“赶牛入栏”一样，需要大量的个人努力。不仅需要提示人工智能，还需要检查其输出，因为有时候看起来可行的部分实际上存在问题。这种体验比他自己直接解决问题更为耗神。

So like it, but that's the currency to be hot. I wonder if there's a phase shift that happens, towards no longer feels like hurting cats, and maybe you'll surprise us how quickly that comes. I believe so. So in formalization, I mentioned before that it takes 10 times longer to formalize a proof that I had. With these modern AI tools, it's also just better tooling. The lean developers are doing a great job adding more and more features and making it user-friendly. It's going from 9 to 8 to 7, okay, no big deal. But one day you'll drop all the one, and that's the phase shift, because suddenly it makes sense when you write a paper to divide it in lean first, or through a conversation with AI, which is generating lean on the fire with you. It becomes natural for journals to accept, and maybe they'll offer expedite refereeing. If a paper has already been formalized in lean, they'll just have to referee to comment on the significance of the results and how it connects to the literature and not worry so much about the correctness, because that's been certified.

所以，就像这样，但这就是热门的趋势。我想知道是否会出现一个阶段性的转变，让事情不再像是在伤害猫一样（意指很难管理），也许你会让我们惊讶于这种转变来得多么迅速。我相信如此。在正规化方面，我之前提到过，把一个证明正规化需要花费原来十倍的时间。而现在有了这些现代AI工具，这一过程变得更加高效。Lean（一个形式验证工具）的开发人员做得很出色，添加了越来越多的功能，使得它更加用户友好。这可能是在逐渐从9到8再到7（即效率逐步提高），没什么大不了的。但有一天你会突然把所有1都去掉，那就是一个阶段性的转变。突然之间，在撰写论文时，先在Lean里细分或者通过与AI的对话来生成Lean（形式化证明）就变得合情合理。这种变化会让期刊自然接受，或甚至提供加快审稿的服务。如果一篇论文已经在Lean中进行了正规化，他们只需要评价结果的重要性以及它与现有文献的关联性，而不必过于担心其正确性，因为那已经得到了认证。

Papers are getting longer and longer in mathematics, and it's harder and harder to get good refereeing for the really long ones, unless they're really important. It is actually an issue, which in the formalization is coming in just the right time for this to be. And the easier and easier to guess, because of the tooling and all the other factors, then you're going to see much more like math, label, grow, potentially exponentially. It's a virtuous cycle, okay. I mean, one phase shift of this type that happened in the past was the adoption latex. So latex is a type of language that all mathematicians use now. So in the past, people used all kinds of word processes and typewriters and whatever. But at some point, latex became easier to use than or other competitors, and that people would switch within a few years, like it was just a dramatic spaceship.

数学论文越来越长，对于那些真正很长的论文来说，很难找到优秀的审稿人，除非它们真的很重要。这实际上是一个问题，而形式化（formalization）正好在这个时候出现，为此提供了解决方案。由于各种工具和其他因素的影响，推断变得越来越容易，因此你会看到数学领域的标记会迅速增加，可能呈指数增长。这是一个良性循环。过去发生过的一个类似的转变是LaTeX的普及。LaTeX是一种现在所有数学家都使用的编程语言。在过去，人们使用各种文字处理工具、打字机等，但在某个时刻，LaTeX变得比其他竞争者更容易使用，于是人们在短短几年内就迅速转向使用LaTeX，这简直是一种戏剧性的转变。

It's a wild out there question. But what year, how far away are we from a AI system being a collaborator on a proof that wins the field's model, so that level? Okay. Well, it depends on the level of collaboration. Yeah. No, like it deserves to be to get the fields model. Like, so half an hour. Already, like, I can imagine if it was a metal-witting paper having some AI systems inviting it. Just, you know, like they all complete alone. It's a very, I use it like it speeds up my own writing. Like, you know, you can have a theorem, you have a proof of three cases, and I write down the proof of first case, and the autocomplete just suggests that now these are the proof of second case of good work. And like, it was exactly correct. That was great. Save me like five to minutes of typing.

这是一个很大胆的问题。不过，我们距离 AI 系统成为一种能够合作完成获得菲尔兹奖难度级别的证明的合作者，还有多少年？好的，这要看合作的具体程度。对，我的意思是，AI 应该能够为获得菲尔兹奖作出足够的贡献。就像半小时内之类的情况。我已经可以想象，有些获奖论文可能会有 AI 系统的参与。比如，它们能够独立完成整个过程。这种体验非常棒，我自己用的时候感觉它可以加快我的写作速度。比如，你可以有一个定理，需要证明三个情况，我写完第一个情况的证明后，自动补全就给我建议了第二种情况的证明，而且完全正确。这非常棒，帮我省下了五到十分钟的输入时间。

But in that case, the AI system doesn't get the fields model. No. I was talking 20 years, 50 years, 100 years. What do you think? Okay. So I gave a bit of an imprint of us about 20, 26, which is now next year. There will be mathematical operations with AI. So not fields model winning, but like actual research level, like published ideas that are in our generation by AI. Maybe not the ideas, but at least some of the computations, the verifications here. I mean, there's that already happened. Yeah. There are problems that were solved by a complicated process, conversing with AI to propose things, and the human goes and tries it, and the contract doesn't work. But the, my, it was a different idea. It's hard to disentangle exactly. There are certainly math results, which could only have been accomplished because there was a method of human authentication and an AI involved.

但在这种情况下，AI 系统并没有获得菲尔兹奖。不是的。我讨论的是 20 年、50 年、100 年以后的情况。你怎么看？好的。我在 2026 年，也就是明年，提到了一些我们的印迹。届时，AI 将参与数学操作。虽然不是获奖级别的工作，但会达到实际研究水平，比如通过 AI 产生的发表理念。或许不是新的概念，但至少会有些计算和验证是这样的。我的意思是，这种事情已经发生了。通过与 AI 的复杂互动，已经有一些问题得到了解决，人类会尝试这些建议，虽然有时候结果不如预期，但其中确实包含了不同的想法。这很难完全理清。毫无疑问，有些数学成果只能通过人类验证和 AI 的结合才得以实现。

But it's hard to sort of disentangle credit. I mean, these tools, they do not replicate all the skills needed to mathematics, but they can replicate sort of some non-trivial percentage of them, you know, 30, 40 percent. So they can fill in gaps. So coding is a good example. So I am, it's annoying for me to code and Python. I'm not a native, no professional programmer. But with AI, the fiction cost of doing it is much reduced. So it fills in that gap for me.

要分清谁的功劳有点困难。我的意思是，这些工具不能完全取代完成数学所需的所有技能，但它们可以复制其中一些不太简单的部分，比如30%到40%。因此，它们能够填补一些空白。编程就是一个很好的例子。对我来说，用Python编程很麻烦，我不是专业程序员。但有了人工智能，编程的难度降低了很多，它帮助我填补了这方面的空白。

AI is getting quite good at literature review. I mean, there's still a problem with hallucinating, you know, references that don't exist. But this I think is a several problems. If you train in the right way and so forth, you can, and verify using the internet. You know, you should, in a few years, get the point where you have a lemma that you need and say, anyone proving the slumber before and they will do basically a fancy web search AI system.

人工智能在文献综述方面变得相当出色。我是说，目前仍然存在一些问题，比如凭空捏造一些不存在的参考文献。但我认为这其中有几个问题。如果以正确的方式进行训练，并通过互联网进行验证，未来几年内，当你需要一个引理时，你就可以问有人以前证明过这个吗？这时人工智能系统基本上会执行一个高级的网络搜索。

So yeah, there are these six papers where something similar has happened. I mean, you can ask you right now and it will give you six papers of which maybe one is legitimate and relevant. One exists but is not relevant for hallucinating. It has a non-zero success rate right now, but there's so much garbage, so much the signal noise ratio is so poor that it's, it's most helpful when you already somewhat know the literature.

所以，是的，有六篇论文出现了类似的情况。我是说，你现在可以去查一下，会找到六篇论文，其中可能只有一篇是合适且相关的。有一篇是存在的，但不相关，因为内容有些不着边际。目前这种情况的成功率虽然不是零，但却有很多垃圾信息，信号噪音比非常低。因此，当你对相关文献已有一定了解时，查阅这些论文才会更有帮助。

And you just need to be prompted to be reminded of a paper that was really subconsciously in your memory. Or it's just helping you discover new you were not even aware of, but is the correct citation. Yeah, that's, yeah, that it can sometimes do, but when it does, it's buried in a list of options to which the other are bad. Yeah, I mean, being able to automatically generate a related work section that is correct. Yeah. That's actually a beautiful thing that might be another phase shift because it assigns quite a correctly.

当你被提示时，你可能会想起一篇其实已在潜意识中记住的论文。或者这个提示可以帮助你发现一些你之前没有意识到但却是正确引用的内容。对，这确实是它可以做到的，但有时候这些正确的引用会被埋在一堆糟糕的选项中。能够自动生成一个正确的相关工作部分是一件很美好的事情。这可能会是另一个重要的跃变，因为它分配得非常准确。

Yeah. It does, it breaks you out of the silos of, yeah, yeah, yeah, yeah, no, there's a big hump to overcome right now. I mean, it's like self-driving cars. The safety margin has to be really high to be feasible. So yeah, so there's a last mile problem with a lot of AI applications that they can do, they can do tools that work 20%, 80% of the time, but it's still not good enough, and in fact, even worse than good, some ways.

是的，它确实把你从孤立状态中解放出来。是的，现在确实有一个需要克服的大障碍。我是说，就像自动驾驶汽车一样，它的安全裕度必须非常高才可行。所以很多人工智能应用都有“最后一公里”问题，它们可以在20%到80%的情况下正常工作，但这仍然不够好，甚至在某些方面比完全运作还要糟糕。

I mean, another way of asking the Fields-Model question is, what year do you think you'll wake up and be like real surprised? You read the headline, the news or something happened that AI did, like, you know, real breakthrough, something. It doesn't, you know, like, Fields-Model, even hypothesis, it could be like really just, this Alpha Zero moment would go that way. Right.

换句话说，Fields-Model问题可以这样问：你觉得在哪一年你会醒来，然后看到新闻头条或者其他消息，说人工智能取得了真正的突破，让你感到很惊讶？这并不一定是关于Fields-Model，甚至只是一个假设，可能会像AlphaZero的时刻那样发展。对吧。

Yeah, this decade, I can see it like making a conjecture between two unrelated, two things that people thought was unrelated. Oh, interesting. Generating a conjecture, that's a beautiful conjecture. Yeah, and actually, it has a real time, so being correct and then meaningful. Because that's actually kind of doable, I suppose, but the word of the data is, yeah, yeah, no, that would be truly amazing.

是的，在这个十年里，我可以将它视作是在两个看似无关的事物之间进行一种推测。哦，真有趣。生成一个推测，这是一个很美的推测。是的，而且实际上，它有一个真实的时间点，因此既正确又有意义。我想这实际上是可行的，但如果能用数据来实现的话，那就太令人惊叹了。

It's kind of a model struggle a lot. I mean, so a version of this is, I mean, the physicists have a dream of getting the AI to discover new laws of physics. The dreams you just feed it all this data, okay? And it says, he was a new patent that we didn't see before. But it actually even struggle, the current state of the art, even struggles to discover all laws of physics from the data.

这有点像一个模型的挑战。我指的是，物理学家们梦想着让人工智能发现新的物理定律。梦想是你只要给它提供所有的数据，然后它就能说出一些我们之前没有看到的新规律。但实际上，即便是最先进的技术，在从数据中发现已有的物理定律方面还是会遇到困难。

Or if it does, there's a big concern with contamination that it did only because, like, it's somewhere in a training data, it did some new, you know, boils, whatever, if you're trying to reconstruct. Part of it is that we don't have the right type of training data for this. Yeah, so for laws of physics, we don't have like a million different universes, we have a million different balls of nature.

或者如果真的出现这种情况，人们就会非常担心污染问题。因为很可能仅仅是因为在某个训练数据中存在类似的内容，导致出现了一些新的东西，比如说一些新的沸腾现象，无论在你试图重建什么的时候。部分原因是我们没有合适类型的训练数据。是的，对于物理定律来说，我们并没有上百万个不同的宇宙，我们只有上百万个自然界的例子。

And a lot of what we're missing in math is actually the negative space of, so we have published things of things that people have been able to prove and conjectures that end up being verified or we can't examples produced. But we don't have data on things that were proposed and they're kind of a good thing to try. But then people quickly realized that it was the wrong conjecture and then they disinnovated, but we should actually change our claim to modify it in this way to actually make it more plausible.

在数学中，我们实际上缺少的很多东西是负空间。我们有已发表的、被证明的东西，也有最终被验证的猜想，或者我们无法举出例子的情况。但我们没有关于那些被提出但很快就被认为是错误猜想的信息。尽管这些猜想最初可能是很好的尝试，但人们很快意识到它们是错误的，然后放弃了。实际上，我们应该改变我们的主张，以这种方式修改它，使其更为合理。

There's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record because it's embarrassing. We make mistakes and we only like to publish our wins. And yeah, it has no access to data to train on. I sometimes joke that basically, you know, I just get AI has to go through a grad school and actually, you know, go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes and learn from that.

有一个试错过程，这是人类数学发现中真正不可或缺的一部分，但我们往往不记录这个过程，因为它让人尴尬。我们会犯错，只喜欢公布我们的成功。而且，AI没有数据可以用来训练。我有时开玩笑说，AI基本上就像是要经历一个研究生阶段，实际上去上研究生课程、完成作业、参加辅导时间、犯错、得到纠正错误的建议，并从中学习。

Let me ask you, if I may, about Gregory Perlman. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you've really fallen love with the problem and it really cannot rest until you solve it. But you also hastened to add that sometimes this approach actually can be very successful. An example you gave is Gregory Perlman who proved the Poincaré Conjecture and did so by working alone for seven years with basically little contact with the outside world. Can you explain this one millennial prize problem that's been solved Poincaré Conjecture and maybe speak to the journey that Gregory Perlman's been on?

让我问一下，如果可以的话，关于格里戈里·佩雷尔曼的问题。你提到过，你在工作中尽量小心，不让一个问题完全占据你的心思。然而，你确实会非常热爱这些问题，直到解决它们为止。但是，你也很快补充说，有时候这种方法实际上会非常成功。你举的一个例子就是格里戈里·佩雷尔曼，他通过独自工作七年、几乎与外界没有联系的方式，证明了庞加莱猜想。你能解释一下这个被解决的千禧年大奖难题——庞加莱猜想，并谈谈格里戈里·佩雷尔曼的这个旅程吗？

All right, so it's a question about curved spaces. That's a good example. I think it was a 2D surface. I'm just assuming round you could maybe be a torus with a hole in it or it can have many holes. And there are many different topologies, a priori that the surface could have. Even if you assume that it's bounded and smooth and so forth. We have figured out how to classify surfaces. As a first approximation, everything is determined. I don't know the genus. How many holes it has. So the sphere has a genus zero. Don't know it has a genus one and so forth.

好的，这个问题是关于曲面空间的。这是一个很好的例子。我想它是一个二维的表面。我假设它是一个圆形的，但它也可能是一个中间有洞的圆环，或者可以有很多孔洞。实际上，这个表面可以有很多不同的拓扑结构。即使假设它是有界的、平滑的等等。我们已经找到了分类曲面的方法。作为一个初步的近似，一切都是决定性的。我不知道它的属（即有多少个孔洞）。例如，球面属为零，不知道的情况下，它可能有一个属是一等等。

And one way you can tell the surface is apart, probably the sphere has which is constantly connected. If you take any closed loop on the sphere like a big close to a rope, you can contract it to a point and while staying on the surface. And the sphere has this property. But a torus doesn't. If you're on the torus and you take a rope that goes around, say the outer diameter, there's no way it can't get through the hole. There's no way to contract at all point.

有一种方法可以判断一个表面是否连通，比如一个球体的表面是完全连通的。如果你在球体上画一个闭合的循环，比如一根大圆圈的绳子，你可以将它收缩到一个点，同时保持在表面上。球体具有这种特性，但环面（甜甜圈形状的物体）则没有。如果你在环面上，用一根绳子围绕它的外直径，就没办法把绳子从中间那个空洞中通过，无法将绳子收缩到一个点。

So it turns out that the sphere is the only surface with this property of contractability up to like continuous deformations of the sphere. So some things that are what are called topologically. So point where you ask the same question, higher dimensions. So it becomes hard to visualize because the surface you can think of as embedded in three dimensions. But as curved free space, we don't have good intuition of four years face to live. And then there are also three spaces that can't even fit into four dimensions. You need five or six or four or higher.

原来，球面是唯一一种具有这种可收缩性质的曲面，能够通过连续变形来进行变换。从拓扑学的角度看，一些问题在更高维度上会变得复杂，因为我们习惯于在三维空间中思考表面。然而，我们对于四维空间的直观感受并不好。此外，还有一些三维空间甚至无法嵌入四维空间，必须至少需要五维、六维或更高维度的空间。

But anyway, mathematically you can still pause this question. That if you have a bounded three dimensional space now, which is also has this simply connected property that every loop can be contracted, can you turn it into a three dimensional version of the sphere. And so this is the point where you can actually weirdly in higher dimensions four and five, it was actually easier. So it was solved first in higher dimensions. There's somehow more room to do the deformation. It's easier to move things around to your sphere. But three was really hard.

无论如何，从数学上来说，你仍然可以提出这个问题：如果你有一个有限的三维空间，并且这个空间具有简单连通的性质——也就是说每一个闭环都可以被收缩，那么你能否将其变形成一个三维版本的球体。这就是问题的奇妙之处：在更高维度的四维和五维空间中，实际上更容易解决这个问题，所以高维的情况先被解决了。在更高维度中，有更多的空间来进行变形，更容易将东西移动到你的球体上。但是在三维空间中，这个问题非常困难。

So people tried many approaches. There's sort of commentary approaches where you chop up the surface into little triangles or tetrahedrons. You just try to argue based on how the faces interact each other. There were algebraic approaches. There's various algebraic objects like the fundamental group that you can attach to these homology and homology and all these very fancy tools. They also didn't quite work. But Richard Hamilton proposed a partial differential equations approach.

人们尝试了许多方法。有一种是“评注式”方法，其中将表面分成小三角形或四面体，然后根据这些面的相互作用来进行推理。还有代数方法，涉及一些代数对象，比如可以与同调、同调群相关联的基本群，以及其他非常高级的工具。然而，这些方法都不太奏效。后来，理查德·汉密尔顿提出了一种基于偏微分方程的方法。

So you take you take so the problem is that you have this object which is sort of secret is a sphere. But it's given to you in a really weird way. So I think of a ball that's been kind of crumpled up and twisted and it's not obvious that it's the ball. But like if you have some sort of surface which is which is a deformed sphere, you could, for example, think of it as a surface of a balloon. You could try to inflate it. You pull it up. And naturally as you fill the air, the wrinkles were sort of smooth out and it will turn into a nice round sphere.

所以问题在于，你有一个物体，它有点像一个秘密的球体。但是，这个球体以一种非常奇怪的方式呈现给你。我把它想象成一个被揉皱和扭曲的球体，以至于看不出它是个球。不过，如果你有一个变形的球体表面，比如想象成一个气球的表面，你可以试着给它充气。随着你充入空气，皱褶自然会变平滑，最终变成一个漂亮的圆球。

Unless of course it was a toy or something like that which is it would get stuck at some point. Like if you inflate a toy it would, there would be a point in the middle when the inner ring shrinks to zero you get a singularity and you can't pull up any further. You can't pull up any further. So if you created this flow which is called Richie flow, which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder to make it look like a sphere.

当然，除非这是一个玩具或类似的东西，那么在某个点它会卡住。比如说，如果你给一个玩具充气，那么在中心会有一个点，内环会缩小到零，形成一个奇点，这时你就无法再继续充气了。所以，如果你创建了一种被称为里奇流的流动方式，它是一种通过不断平滑任意表面或空间，使其变得越来越圆，最终看起来像一个球体的方法。

And he wanted to show that either this process would give you a sphere or it would create a singularity. I can very much like how PDs either have global irregularity or finite and blow up. Basically it's almost exactly the same thing. It's all connected. And so and he showed that for two dimensions, two initial surfaces, if you start with some clear neck, no singularities ever formed. You never manage a trouble and you could flow and it will give you a sphere.

他想要展示的是，这个过程要么会给你一个球体，要么会产生一个奇点。我非常喜欢类似偏微分方程的问题，它们要么具有全局的不规则性，要么是有限的并且会在某处爆炸。实际上，这几乎是完全相同的概念，这一切都是相互关联的。他还证明，对于二维的情况，如果从两个初始曲面开始，并且初始形状足够清晰，没有奇点会产生，你就不会遇到麻烦，可以继续进行流动并最终得到一个球体。

And so he got a new proof of the two dimensional results. But whether that's a beautiful explanation of a Richie flow and its application and its context, how difficult is the mathematics here for the 2D case? Yeah, these are quite sophisticated equations. On par with the Einstein equations. Okay. It's slightly simpler but yeah, but they will consider hard nonlinear equations to solve. And there's lots of special tricks in 2D that that helped. But in 3D, the problem was that this equation was super critical. It has the same problem as Navier's dogs. As you blow up, maybe the curvature could get constrained in finite small and small regions. And it looked more and more nonlinear and things just look worse and worse.

因此，他获得了一个关于二维结果的新证明。但是，关于这是一个关于Ricci流的美丽解释及其应用和背景，在二维情况下，数学有多难呢？是的，这些方程是相当复杂的。可以与爱因斯坦方程相提并论。好吧，尽管它稍微简单一些，但仍然被认为是难以解决的非线性方程。在二维中，有很多特殊技巧可以帮助解决问题。但是在三维中，问题在于这个方程是超临界的。它与纳维-斯托克斯方程有同样的问题。当你放大时，曲率可能会在有限的更小和更小的区域内受到约束，方程看起来越来越非线性，情况也越来越糟。

And we all kinds of singularities that showed up. Some singularities, there's these things called neck pinches where the surface creates a big hip like a barbell and it pinches at a point. Some singularities are simple enough that you can sort of see what you do next. You just make a snip and then you can turn one surface into two and you built them separately. But there was the prospect that from really nasty, like, notive singularities showed up that you couldn't see how to result in any way, that you couldn't do any surgery too. So you need to classify all the singularities. What are all the possible ways things can go wrong?

我们遇到各种奇点。其中一些奇点称为"颈缩"，表现为表面形成像哑铃一样的形状，并在一点处收缩。有些奇点相对简单，你可以看出接下来该怎么做。你只需进行一个切割，就能将一个表面分成两个，并分别进行处理。然而，也有可能出现非常复杂、棘手的不规则奇点，你看不出如何解决，也无法进行任何切割手术。因此，我们需要对所有奇点进行分类，找出所有可能出问题的方式。

So what Perman did, first of all, he made the problem, he turned the problem super critical problem to a critical problem. I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics. So he introduced something which is now called Perman's reduced volume and Perman's entropy. He introduced new quantities, kind of like energy, that looked the same at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before. And then he had to solve, he still had to analyze the singularities of this critical problem.

所以，首先，佩雷尔曼采取的做法是，他将这个极为复杂的问题转化为一个临界问题。正如我之前提到的，能量（哈密顿量）的引入确实澄清了牛顿力学。因此，佩雷尔曼引入了一些现在被称为佩雷尔曼化简体积和佩雷尔曼熵的东西。他引入了一些类似于能量的新量，这些量在每个尺度上看起来都是一样的，这将问题转化为一个临界问题，使得非线性部分看起来没有之前那么可怕。然后，他仍然需要分析这个临界问题中的奇点。

And that itself was a problem similar to this weight-capsing at work on actually. So on the level of difficulty of that, so he managed to classify all the singularities of this problem and show how to apply surgery to each of these and through that was able to result the point in Jack here. So quite like a lot of really ambitious steps and like nothing that a large language model today, for example, could I mean, at best, I could imagine a model proposing this idea as one of hundreds of different things to try. But the other 99 would be complete dead ends, but you don't only find out after months of work. He must have had some sense that this was the right tractable suit because it takes years to get them pump A to B.

这本身就是一个类似于他在工作中遇到的问题，涉及到某种“权重限制”的挑战。在处理这种难度的问题上，他成功地对这个问题的所有奇点进行分类，并展示了如何对每个奇点进行“手术”处理。通过这种方法，他在某种程度上解决了杰克这里的关键问题。他采取了许多雄心勃勃的步骤，这些步骤是目前的大型语言模型无法完全实现的。一个模型或许可以将这作为成百上千个尝试中的一个，但其他99个尝试可能最终证明是完全无效的，而你只有经过数月的努力后才能发现。他一定对这条可行的路径有某种明确的感觉，因为这需要多年才能将一个想法从A点推进到B点。

So you've done like you said, actually, you see even strictly mathematically, but more broadly in terms of the process, he done similarly difficult things. What can you infer from the process he was going through because he was doing it alone? What are some low points in a process like that? When you start to like, you've mentioned heart share like AI doesn't know when it's failing. What happens to you? You're sitting in your office when you realize the thing you did the last few days, maybe weeks is a failure. Well, for me, I switch to a different problem. So I'm a fox, I'm not a hedgehog, but you will generally, that is a break that you can take is to step away and look at it in a problem. You can modify the problem too. You can ask them to cheat.

所以，你确实按照自己所说的去做了，实际上，不仅是严格的数学方面，更广泛地说从整个过程来看，他也完成了类似艰难的事情。你能从他独自经历的过程中推断出什么？这样的过程中会有哪些低谷？你提到过心累，就像AI不知道自己什么时候失败了。当你坐在办公室里意识到过去几天甚至几周的努力是失败的时，你会有什么感受？对我来说，我会转向解决另一个问题。所以我是狐狸，而不是刺猬，不过一般来说，你可以暂时离开一下，重新审视问题。你还可以修改问题，甚至请人帮忙想办法。

If there's a specific thing that's blocking you, that just some bad case keeps showing up for which your tool doesn't work, you can just assume by fear, this bad case doesn't occur. You do some magical thinking, but strategically, okay, for the point to see if the rest of the argument goes through. If there's multiple problems with your approach, then maybe you just give up. If this is the only problem that we know, then everything else checks out, then it's still worth fighting. So yeah, you have to do some so-ford reconnaissance sometimes. That is sometimes productive. Just assume we'll figure it out. Sometimes actually it's even productive to make mistakes.

如果有某个特定的问题阻碍了你，比如你的工具在某些糟糕的情况下不起作用，你可以先假设这种糟糕情况不会发生。你可以进行一些有策略的“神奇思考”，看看这样是否能让其余的论证通畅。如果你的方法有多个问题，也许你就会放弃。但如果这只是我们知道的唯一的问题，而其他部分都没问题，那么还是值得继续努力。所以，有时候你需要做一些预先的侦查，这可能会有益处。假设我们会找到解决办法，有时甚至在出错过程中也能有所收获。

So one of the, I mean, there's a project which actually we want some prizes for. We worked on this PD problem, again, actually this blow-off regularity type problem. It was considered very hard. John McCain, another few of his methods, he worked on a special case of this, but he could not solve the general case. We worked on this problem for two months and we thought we solved it. We had this dispute argument that I think fit and we were excited. We were planning celebration, we will get together and have champagne or something.

我们有一个项目，实际上我们因此赢得了一些奖项。我们研究的是一个偏微分方程的问题，也就是所谓的爆破规律性问题。这个问题一直被视为非常困难。约翰·麦凯恩和他的几个同事曾经研究过这个问题的一种特殊情况，但他们没能解决一般情况下的问题。我们花了两个月的时间研究这个问题，以为已经解决了。我们找到了一种看似合适的推理方法，非常激动，甚至还计划庆祝一下，聚在一起开香槟什么的。

We started riding it up and one of us, not me, I could, but another call of a, said, oh, in this lemma here, we have to estimate these 13 terms that show up in this expansion. It's made 12 of them, but in our notes, I can't find it for the estimation of 13th, can you? Someone supply that. I said sure, look at this and I we didn't cover that we completely omitted this term. This term, we worse than the other 12 terms put together. In fact, we could not estimate this term. We tried for a few more months and all different permutations and there was always this one thing that we could not control.

我们开始讨论这个问题，其中一位同事（不是我，但我也能说）提到，在这个引理中，我们需要估算在展开式中出现的13个项。我们的笔记中已经处理了其中的12项，但对于第13项的估算部分找不到相关内容，你能找到吗？有人说可以，并让我看看他的笔记。这时我才发现，我们完全忽略了这个项。这个项的复杂程度比其他12项加起来都要棘手。事实上，我们无法估算这个项。我们尝试了几个月，尝试了各种方法组合，但始终有一个问题我们无法控制。

This was very frustrating, but because we had already invested months and months of effort and was already, we stuck at this. We tried increasingly desperate things and crazy things. After two years, we found that the picture is somewhat different, but quite a bit from our initial strategy, which didn't generate these permutations and actually solve the problem. We solved the problem after two years, but if we hadn't had that initial full-storm of nearly solving the problem, we would have given up by month two or something and worked on an easier problem.

这件事让人非常沮丧，但因为我们已经投入了几个月甚至几个月的努力，所以我们坚持了下来。我们尝试了越来越多绝望的和疯狂的方法。两年后，我们发现情况有些不同，但与我们最初的策略有很大差别。最初的策略并没有产生这些变化，也没有真正解决问题。经过两年时间，我们终于解决了问题，但如果我们当初没有几乎解决问题的强烈信念，我们大概会在第二个月左右放弃，然后去解决一个更简单的问题。

If we had known it would take two years, not sure we would have started the project. Sometimes actually having the asylambus in Newark, they had an incorrect version of the measurement of the size of the earth. He thought he was going to find a new trade route in India. At least that was how he sold it in his perspective. It could be that he secretly knew. Just on the psychological element, do you have emotional or self-doubt, the just overwhelmed you, and things like that? This stuff feels like math is so engrossing that it can break you. When you invest so much yourself in the problem and then it turns out wrong, you could start to...

如果我们知道需要两年时间，我们可能就不会开始这个项目。有时候，就像当时在纽瓦克，他们对地球大小的测量有误。有人以为自己会找到一条通往印度的新贸易路线，至少在他看来是这样。他可能暗地里知道真相。在心理层面上，是否有情绪或自我怀疑不断困扰你，让你感到不知所措之类的？这些事情让数学显得如此吸引人，以至于它可能会让你崩溃。当你投入大量心力在问题上，而结果却是错的，你可能会开始……

Similar way chess has broken some people. I think different mathematicians have different levels of emotional investment in what they do. I think for some people it's as a job. You have a problem and if it doesn't work out, you will be on the next one. The fact that you can always move on to another problem, it reduces the emotional connection. There are cases, so there are certain problems that are what about that go diseases, where just latch on to the one problem and they spend years and years thinking about nothing but that one problem. Maybe the career suffers and so forth.

类似于国际象棋让一些人崩溃，我认为不同的数学家对他们所做的事情有不同的情感投入程度。对有些人来说，数学只是一项工作。你有一个问题要解决，如果这个问题没有结果，你可以继续处理下一个问题。能够总是转向下一个问题，这种情况减少了情感上的依赖。然而，也有一些问题犹如难以摆脱的顽疾，让人久久不能放手，他们可能会花上多年时间，只专注于这个问题，也许他们的职业生涯因此受到影响。

I could get this big win this world. Once I finish this problem, I will make up for all the years of lost opportunity. Occasionally it works. I really don't recommend it for people who are far too right-forward to you. I've never been super invested in any one problem. One thing that helps is that we don't need to call our problems in advance. When we do grab proposals, we want to study this set of problems. But even then we don't promise, definitely by five years, I will supply a proof of all these things.

我可以在这个世界上获得巨大的成功。一旦我解决了这个问题，我就能弥补过去多年的机会损失。有时候，这种方法奏效。但我真的不建议那些性格过于直接的人使用。我从来没有对某一个问题投入过超级多。一个有帮助的事情是，我们不需要提前确定要解决哪些问题。即使我们抓住了几个方案，我们也会想要研究这些问题。但即便如此，我们也不会保证在五年内一定会提供所有这些问题的证明。

You promise to make some progress or discover some interesting phenomena and maybe you don't solve the problem, but you find some related problem that you can say something new about and that's a much more feasible task. But I'm sure for you there's problems like this. You have made so much progress towards the hardest problems in the history of mathematics. Is there a problem that just haunts you? It sits there in the dark corners. Twin prime conjecture, Raymond hypothesis, go luck conjecture. Twin prime, that's...

你承诺会取得一些进展或发现一些有趣的现象，也许你不能解决这个问题，但你发现了一些相关的问题，可以提出新的见解，这是一项更可行的任务。我相信对你而言，也有类似的问题。你在数学史上一些最难的问题上取得了很大进展。那么，有没有一个问题让你感到挥之不去？它就像藏在阴暗角落里的幽灵。比如孪生素数猜想、黎曼假设、哥德巴赫猜想。孪生素数，那就是...

Again, the problem is that the Raymond hypothesis is so far out of reach. I think so. Yeah, there's no even viable strategy. Even if I activate all the cheats that I know of in this problem, there's still no way to get me to be. I think it needs a breakthrough in another area of mathematics to happen first. For someone who recognized that it would be a useful thing to transport into this problem. We should maybe step back for a little bit and just talk about prime numbers.

再次，这个问题在于雷蒙德假说目前仍然遥不可及。我认为确实是这样。是的，甚至没有一个可行的策略。即使我在这个问题上使用了我知道的所有“作弊”技巧，依然无法解决。我认为首先需要在数学的其他领域有所突破，然后才能对这个问题有所助益。我们也许应该退一步，先来聊聊质数的问题。

So they're often referred to as the atoms of mathematics. Can you just speak to the structure that these atoms... So the natural numbers have two basic operations, attention on addition and multiplication. If you want to generate the natural numbers, you can do one or two things. You can just start with one and add one to itself over and over again and that generates you the natural numbers. So additively, they're very easy to generate. One, two, three, five. Or you can take the prime number, if you want to generate multiplicatively, you can take all the prime numbers, two, three, five, seven and multiply them all together. Together they kiss you all either, the natural numbers, except maybe four, one. So there are these two separate ways of thinking about the natural numbers. I'm adding to point of view and a multiplicative point of view. Separately, they're not so bad. Any question about that natural numbers, only was addition, as well as easier to solve.

它们通常被称为数学的原子。关于这些原子的结构，简单来说，自然数有两个基本运算：加法和乘法。如果你想生成自然数，你可以有两种做法：可以从1开始，不断给自己加1，这样就生成了自然数。所以从加法的角度看，它们很容易生成：1、2、3、5。或者，如果你想用乘法来生成，可以从质数开始，比如2、3、5、7，把它们互相乘起来。这样几乎生成了所有自然数，除了可能的4和1。所以，关于自然数，有两种不同的思维方式：加法的视角和乘法的视角。单独来看，两者都不难。而关于自然数的问题，通常用加法方式来解决更加容易。

Any question that only was multiplication is a little bit easier to solve. But what has been frustrating is that you combine the two together. And suddenly, you get an extremely rich, I mean, we know that there were statements in numbers that are actually as undesirable. There are certain polynomials in some number variables. You know, it's the solution in the natural numbers and the asset depends on an undecidable statement, like whether the axioms of mathematics are consistent or not. But yeah, but even the simplest problems that combine something more multiplicative such as the primes with some additives such as chipping by two. Separately, we understand both from well, but if you ask when you shift the prime by two, do you, can you get up, how often can you get another prime? It's been amazingly hard to relate the two.

任何只涉及乘法的问题通常会比较容易解决。但是令人感到沮丧的是，当你把乘法和另一种操作结合在一起时，问题就变得非常复杂。我们知道在数字中有一些命题实际上是不可判定的，有些关于多个变元的多项式的问题，其解在自然数中存在与否可能取决于某个不可判定的命题，比如数学公理是否一致。但即便是那些看似简单却结合了乘法与其他运算的问题，比如以素数为基础，加上类似加2这样的简单操作，单独来看，我们对两者各自都比较了解。但如果问你把一个素数加2后，能得到另一个素数的频率是多少，这两个概念结合在一起后，要理解它们的关系就变得非常困难。

And we should say that the twin prime conjectures just that it posits that there are infinitely many pairs of prime numbers that differ by two. Now, the interesting thing is that you have been very successful at pushing forward the field in answering these complicated questions of this variety. Like you mentioned, the green tile theorem, it proves that prime numbers contain arithmetic progressions of any length. Which is mind-blowing, you could prove something like that. Right. Yeah, so what we've realized because of this type of research is that there's different patterns have different levels of indestructibility. So what makes the twin prime common hard? You could take all the primes in the world, three, five, seven, eleven, so forth. There are some twins in there, eleven and thirteen is a twin prime pair of twin primes so forth.

我们应该提到，孪生素数猜想就是这样一个命题：它认为有无限多对素数，且它们的差是二。现在，有趣的是，你在解答这类复杂问题的领域取得了很大的成功。就像你提到的格林-陶定理，它证明了素数包含任意长度的等差数列。这实在令人震惊，你竟然能够证明这样的事情。对吧。是的，因此我们通过这类研究意识到，不同的模式有着不同程度的坚固性。那么是什么让孪生素数猜想变得如此困难呢？你可以看看所有的素数，比如三、五、七、十一等等。在这些素数中也存在孪生素数，例如十一和十三就是一对孪生素数。

But you could easily, if you wanted to, redact the primes to get rid of these twins. The twins, they show up and they're infinitely many of them, but that you recently sparse. Initially, it's quite a few, but once you got the millions, trillions, they become rare and rare. And you could actually just, if someone was given access to the database of prime, you just edit it out a few primes here and there, they could make the trim package of your false by just removing like 0.0 or 1% of the primes or something, just well chosen to do this. And so you could present a censored database of the primes, which passes all of the statistical tests of the primes. You know, it obeys things like the problem of theorem and other things about the primes, but doesn't contain any trim primes anymore.

你可以很容易地删除素数中的双生素数。如果你想这样做的话，尽管双生素数在数学上有无限多个出现，但实际上它们是相对稀疏的。在一开始的时候，这些双生素数会挺多，但随着数字增大到百万、万亿，它们就越来越少了。实际上，如果有人能访问素数数据库，只需删除个别素数，即使只是删除0.01%或1%的素数，只要经过精心挑选，就能让这个精简的数据库变得不完整。即便这样处理后，你可以呈现一个被审查过的素数数据库，它可以通过所有素数的统计测试，比如满足定理的要求等，但里面却不再包含任何双生素数。

And this is a real obstacle to the trim prime conjecture. It means that any proof strategy to actually find trim primes in the actual primes must fail when applied to these slightly edited primes. And so it must be some very subtle, delicate feature of the primes that you can't just get from like, like I could get statistical analysis. Okay, so that's all. Yeah. On the other hand, I think progression has turned out to be much more robust. Like you can take the primes and you can eliminate 99% of the primes actually. And you can take any 90% of the one and it turns out and another thing we prove is that you still get asmic progressions. Asmic progressions are much, you know, they're like cockroaches of arbitrary length. Yes. Yes. That's crazy.

这对修剪质数猜想来说是一个真正的障碍。这意味着，任何寻找修剪质数作为实际质数的证明策略，在应用于这些略作编辑的质数时都将失败。因此，这一定是质数中一些非常微妙而精细的特征，而不仅仅是我可以通过统计分析得到的那种特征。另一方面，我认为数列的特性被证明更具韧性。比如，你可以从质数中删去99%的质数，然后从剩下的质数中任意选择90%的部分，我们证明了，你仍然可以得到类似等差数列的结构。这种进展就像蟑螂一样无处不在，并且可以任意延长。是的，是的，这很疯狂。

Yeah. So for people who don't know, I think progression is a sequence of numbers that differ by some fixed amount. Yeah. But it's again, like it's an infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary, that's the progression. You only get quite short progressions. But you're saying twin primes not an infinite monkey of phenomena. I mean, it's a very subtle one. It's still an infinite monkey phenomenon. Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact, the infinite monkey theorem would all be you're saying that twin prime is it doesn't, you can't use the same tools. Like, it doesn't appear random almost.

好的，对于不了解的人，我认为数列是由一些固定差值构成的数字序列。不过，这就像是一个无限猴子定理的现象。对于任何一组固定长度的集合，你都不会得到随意的进程，只能得到相当短的数列。但你提到孪生素数并不是无限猴子现象。我是说，这个现象非常微妙，但它仍然是一个无限猴子现象。如果素数真的完全随机，如果素数是由猴子随机生成的，那么实际上无限猴子定理就能完全解释这一现象。而你所说的孪生素数，你不能用相同的方法去研究。因为它看起来几乎不是随机的。

Well, we don't know. Yeah. We believe the primes behave like a random set. So the reason why we care about the trim and how conjecture is a test case for whether we can genuinely completely say with 0% chance of error that the primes behave like a random set. Okay. Random random versions of the primes we know contain twins, at least with 100% probably, or probably 10 to 100% as you go out further further.

好的，我们不知道。是的。我们相信质数的行为类似于随机集合。因此，我们关心"三元组和"猜想的原因是想测试我们是否可以真正、完全、并且在0%错误率下确定质数的行为像一个随机集合。好的。我们知道，在随机版本的质数中，至少有高达100%或10%到100%的概率包含双胞胎质数，随着范围的扩大，这个概率也会增加。

Yeah. So the primes we believe that the random, the reason why primes are indestructible is that regardless of whether it looks random or looks structured like periodic, in both cases, the arithmetic progressions appear, but for different reasons. And this is basically all the ways in which the thing, there are many proofs of these sort of arithmetic progression after the year and they're all proven by some sort of dichotomy, where your set is either structured or random and in both cases, you can say something and then you put the two together.

好的。所以我们相信质数之所以"坚不可摧"，是因为无论它们看起来是随机的还是有规律的（比如周期性的），在这两种情况下都会出现算术级数，但原因不同。这基本上涵盖了所有算术级数在这一年之后存在的方式，并且这些证明都是通过某种二分法来实现的。也就是说，你的集合要么是有结构的，要么是随机的，但无论哪种情况，你都能得出一些结论，然后把这两个情况结合起来。

But in twin primes, if the primes are random, then you're happy, you win, if the primes are structured, they could be structured in a specific way that eliminates the twins. And we can't rule out that one conspiracy. And yet you're able to make a azanish term progress on the K2Po version. Right.

在孪生素数问题中，如果素数是随机的，那结果令人满意，你成功了。如果素数有特定的结构，那么这种特定的结构可能会导致孪生素数不存在。我们无法排除这种情况。然而，你仍然能够在K2Po版本上取得显著进展。对吧。

Yeah. So the one thing about conspiracies is that any one conspiracy theory is really hard to disprove. That, you know, if you believe the word is what by lizards is a here's some evidence that it's not what I mean, this is what that episode's trying to buy the lizards. Yeah. You may have encountered this kind of phenomenon.

是的，关于阴谋论，有一点是每个阴谋论都很难被彻底否定。比如，如果你相信世界是由蜥蜴人掌控的，我给你一些证据证明不是这样，但这并不是那个节目所试图传达的信息。你可能也遇到过这种现象。

Yeah. So like, I'm a pure like there's there's almost no way to it. Definitely will not I can explain the same as true in mathematics. A conspiracy is totally devoted to learning twin primes. You know, like you would have to also infiltrate other areas of mathematics. But like it could be made consistent at least as far as we know.

好的。可以翻译成：是的，我完全理解你的意思。几乎没有其他解释。这在数学中同样适用。一个致力于学习孪生素数的团体，可能还需要渗透到数学的其他领域。不过，就我们所知，这至少在理论上是可能做到的。

But there's a weird phenomenon that you can make one conspiracy, one conspiracy, rule out other conspiracies. So, you know, if the word is one by this is, it can't also be one by the least. Right. Right. So one unreasonable thing is it's hard to disprove. But more than one, there are there are tools.

有一种奇怪的现象，就是一个阴谋论可以排除其他阴谋论。例如，如果这个世界是被一个特定的阴谋控制的，那么就不可能同时被另一个阴谋支配。一个不合理的事情很难被证伪，但如果有多个不合理的事情存在，就有方法可以进行排除或验证。

So yeah. So for example, we know there's simply many primes that are no two, which are so there is a pair of the pride which differ by at most 246 actually is is the code. So there's a bound. Yes. Right. So like the twin primes, they're called cousin primes that differ by by four. This is called sexy primes that differ by six. What are sexy primes? Primes that differ by six. The name is much less. Of course, there's much less exciting than the name suggests.

好的。举个例子，我们知道有很多质数对，它们之间的差值不会超过246。所以这是一个界限。就像孪生质数一样，有一种质数对被称为"堂兄质数"，它们的差值为4。而差值为6的质数对被称为"性感质数"。实际上，"性感质数"这个名称比它们本身要显得更有趣。

So you can make a conspiracy rule out one of these. But like once you have like 50 of them, it turns out that you can't rule out all of them at once. It just requires too much energy somehow in this conspiracy space. How do you do the bound part? How do you how you develop a bound for the different teen primes?

你可以想出一个阴谋论来排除其中一个。但是，一旦你有大约50个这样的阴谋论，你会发现你无法同时排除所有的。这在某种程度上需要太多的精力。那么，你是如何实现界限部分的呢？你是如何为不同的十几岁素数设定界限的？

Okay. So there's an infinite number of. So it's ultimately based on what's called the pigeonhole principle. So the pigeonhole principle is the statement that if you have a number of pigeons and they all have to go over to the pigeonholes and you have more pigeons than pigeonholes, then one of the pigeonholes has to have at least two pigeons. So that has to be two pigeons that are close together.

好的。所以这是一个无限数量的问题。这个问题最终基于一个被称为“鸽巢原理”的理论。鸽巢原理的意思是，如果你有一定数量的鸽子，它们都需要进入鸽巢，而鸽子的数量多于鸽巢的数量，那么必须至少有一个鸽巢里有两个或更多的鸽子。因此，必然会有两个鸽子靠得很近。

So for instance, if you have a hundred numbers and they all range from one to a thousand, two of them have to be at most 10 apart. Because you can divide up the numbers from one to a hundred into one hundred pigeonholes. Let's say they are 101 numbers. 101 numbers, then two of them have to be distance less than 10 apart because two of them have to belong to the same pigeonhole. So it's a basic basic feature of a basic principle in mathematics.

比如说，如果你有一百个数字，这些数字的范围都在一到一千之间，其中必定有两个数字的差值不超过10。因为你可以把一到一百的数字划分成100个组。如果有101个数字，那么其中有两个数字的差值必定小于10，因为它们必定落在同一个组里。这是数学中一个基本的原理。

So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out. That a few and a few numbers are primes. But it turns out that there's a way to assign weights to numbers. So there are numbers that are almost primes, but they don't have no factors at all other than themselves in one. They have very few factors. And it turns out that we understand almost primes a lot better than those ant primes. So for example, it was known for a long time that they were trying to almost primes. This has been worked out. So almost primes are something we cannot understand. So you can actually restrict the attention to a suitable set of almost primes. And whereas the primes are very sparse overall relative to the almost primes, actually are much less sparse.

所以直接使用质数的问题在于，随着数值的增大，质数之间的间隔越来越大。质数是很少的，只有极少数的数字是质数。但是，有一种方法可以给数字分配权重。于是，有些数字虽然不是完全的质数，但它们只有很少的因数，除了1和它们本身之外几乎没有其他因数。事实证明，我们对"几乎质数"的理解要好于这些完全的质数。例如，早就知道人们在尝试研究"几乎质数"。这方面的研究已经有了一些结果。实际上，我们是可以理解"几乎质数"的。因此，你可以把注意力限制在一个适当的"几乎质数"集合上。相对于质数而言，"几乎质数"实际上要稠密得多。

They make you can set up a set of almost primes where the primes of density likes a 1%. And that gives you a shot at proving by applying some sort of pigeonhole principle that there's pairs of primes that are just only a hundred and a hundred apart. But in order to prove the training ground conjecture, you need to get the density of primes. This is almost up to up to a 50%. Once you get up to 50%, you would get trim primes. But unfortunately, there are barriers. We know that no matter what kind of goods that are almost primes you pick, the density of primes can never get up off 50%.

他们让你可以建立一个几乎为素数的集合，其中的素数密度大约为1%。这给了你一个通过某种鸽笼原理来证明成对素数之间只相隔一百的机会。然而，为了证明训练场猜想，你需要将素数的密度提高到接近50%。一旦达到50%，你就会得到精简素数（trim primes）。但是，不幸的是，目前存在障碍。我们知道，无论你选的几乎素数有多好，其素数密度永远无法达到50%。

It's called the parody barrier. And I would love to find, yes, one of my long-term dreams is to find a way to breach that barrier because it would open up not only to trim out conjecture, the go-back conjecture, and many other problems in number theory are commonly blocked because our current techniques would require going beyond this theoretical parody barrier. It's like pulling past the speed of light.

这被称为“悖论屏障”。我一直希望找到一种方法突破这个屏障，这是我长期以来的梦想之一。因为这样不仅可以解决推测性问题，还能攻克回归猜想以及其他许多数论中的难题。目前，我们的技术常常因为这个理论上的悖论屏障而受阻。就像超越光速一样困难。

Yeah, so we should say a 20-prime conjecture. One of the biggest problems in the history of mathematics, go-back conjecture also. They feel like extra neighbors. Is there been days when you felt you saw the path? Oh, yeah. Yeah, sometimes you try something and it works super well. You again, again, the sense of mathematical smell we talked about earlier. You learn from experience when things are going too well because there are certain difficulties that you sort of have to encounter.

好的，我们应该称之为“20-素数猜想”。这是数学历史上最大的难题之一，还有“回归猜想”。它们就像额外的邻居。有时候你是否觉得自己看到了通往成功的路径？哦，是的。有时候你尝试某些方法，它们效果非常好。正如我们之前所说的数学直觉，通过经验你会学到，当事情进展过于顺利时，你可能会遇到某些困难，这是无法避免的。

I think the way a colleague might put it is that if you are on the streets of New York and you put in a blindfold and you put in a car and after some hours, the blindfolds off and you're in Beijing. That was too easy somehow. There was no ocean being crossed. Even if you don't know exactly what was done, you're suspecting that something wasn't right. But is that still in the back of your head to do you return to the prime numbers every once in a while to see? Yeah, when I have nothing better to do, which is less than that time. It's busy with so many things that you say.

我想一位同事可能会这样说：如果你在纽约的街头，戴上眼罩和坐进一辆车，几个小时后摘掉眼罩发现自己在北京。这个过程似乎太简单了，仿佛根本没有跨越过海洋。即使你不完全明白发生了什么，你也会觉得有些不对劲。但这件事情还会在你脑海中萦绕吗？你会时不时地回到质数这个问题上来看吗？是的，当我没有其他更重要的事情时，我会想到它，不过这样的时间越来越少了，因为总是有太多事情要忙。

But yeah, when I have free time and I'm too frustrated to work on my sort of view, research projects, I also don't want to do my ministry of sub-order. I don't want to do some errands for my family. I can play with these things for fun and usually you get nowhere. Yeah, you have to just say, okay, fine. Once again, nothing happened. I will move on. Very occasionally, one of these problems I actually solved. Sometimes as you say, you think you solved it and then you're forward for maybe 15 minutes and then you think I should check this because this is too easy.

好的，当我有空的时候，如果我对自己的观点和研究项目感到太沮丧，不想做那些的时候，也不想处理家庭琐事，我会玩玩这些东西来娱乐自己。通常情况下，你什么都做不成。是的，你只能说，好吧，又一次，什么也没发生，我会继续前进。偶尔，我能解决其中一个问题。有时候，正如你所说，你觉得你解决了问题，然后兴奋了大约15分钟，接着你会想，我应该检查一下，因为这似乎太简单了。

It could be true and usually is. What's your gut say about when these problems would be solved? When prime and go back to prime, I think we'll keep getting more partial results. It doesn't need at least one, this parity barrier is the biggest remaining obstacle. There are simpler versions of the conjecture where we are getting really close. So I think we will, in 10 years, we will have many more much closer results. We may not have the whole thing.

这可能是真的，而且一般也确实如此。你直觉上觉得这些问题会在什么时候解决？当达到质数状态然后返回质数状态时，我认为我们会不断获得更多的部分结果。并不需要至少有一个，奇偶性障碍是最大的剩余障碍。关于这个猜想，还有一些更简单的版本，我们已经非常接近解决。所以我认为，在未来10年内，我们会取得更多更接近的结果。可能我们不会完全解决这个问题。

So, trend times is somewhere close. Reemnant hypothesis, I have no, I mean, it has happened by accident, I think. So the remnant hypothesis is kind of more general conjecture about the distribution of prime numbers. It's a sort of viewed model plicatively. For questions only involving multiplication, no addition. The primes really do behave well, I as randomly as you could hope. So there's a phenomenon in probably called square cancellation that if you want to pull, say, America upon some issue and you ask one or two voters and you may have sampled a bad sample and then you get a really imprecise measurement of full average.

所以，趋势的时机差不多临近了。残余假设，我没有完全理解，我的意思是，我认为这可能是偶然发生的。因此，残余假设是一种关于素数分布的更一般性的猜想。这种猜想在某种程度上是从乘法的角度来看的，仅涉及乘法的问题，不涉及加法。素数确实表现得很好，尽可能地随机。这儿有一个在概率中称为平方消元的现象。比如说，如果你想对某个议题进行调查，只问一两个选民，你可能会抽到一个不好的样本，结果你得到的对于总体的平均值是非常不准确的。

But if you sample more and more people, the accuracy gets better and better. And it actually improves the square root of the number of people you sample. So if you sample a thousand people, you can get like a two, three percent margin of error. So in the same sense, if you measure the primes in a certain more plicative sense, there's a certain type of statistic you can measure. It's called the remunisator function and it fluctuates up and down. But in some sense, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random. And there's a very precise way to quantify that. And the remunapalysis is a very elegant way that captures this. But as with many other ways in mathematics, we have very few tools to show that something really genuine behaves like Biddy random.

如果你对更多的人进行抽样，准确性会越来越高。实际上，它会以采样人数的平方根比例提高。比如，如果你抽样一千人，误差范围可能在百分之二到百分之三之间。同样地，如果你用某种方式来测量质数，有一种统计量可以用来测量，这被称为"瑞曼函数"（Riemann zeta function），它会上下波动。但从某种意义上来说，随着你对越来越多的数据取平均值，如果你抽样更多，波动幅度应该会像是随机的那样减小。对此，有一种非常精确的量化方法。实际操作中，瑞曼分析是一种优雅的方法，它捕捉了这种波动。但是，就像数学中的许多其他方法一样，我们很少有工具能证明某样东西确实像看上去那样随机。

And this is not just a little bit random, but it's as asked in that behaves as random as a duchyly random set. This square root cancellation. And we know here because of things related to parity from actually most of us, usual techniques cannot hope to settle this question. The proof has to come on a left fuel. But what that is, no one has any serious proposal. And there's various ways to sort of, as I said, you can modify the primes a little bit and you can destroy the remunapalysis. So it has to be very delicate. You kind of find something that has huge margins of error. It has to be just barely work. And there's like all these pitfalls that you have like dodge very adeptly. The prime numbers are just fascinating.

这不仅仅是有一点随机性，而是像一个完全随机的集合一样表现得随机。这种表现被称为平方根消除。由于与奇偶性有关的原因，我们知道大多数常规方法无法解决这个问题。证明必须来自一个意想不到的方向。但是什么方向，目前没有任何具体的提议。正如我之前提到的，有多种方式可以稍微调整素数，但这可能破坏其原有特性。因此，需要非常谨慎地处理，你需要找到一个仅仅能够确保有效的方法，并面临诸多可能的陷阱，需要非常巧妙地避开。素数真是令人着迷。

Yeah. What do you most mysterious about the prime numbers? That's a good question. So like, conjectually, we have a good model of them. I mean, as I said, I mean, they have certain patterns like the primes are usually odd, for instance. But apart from these obvious patterns, they behave very randomly and just assuming that they behave. So there's something called the cream of random model of the primes. But that after a certain point, primes just behave like a random set. And there's various flight modifications as a model. But this has been a very good model. It matches the numerics. It tells us what to predict. Like, I can tell you of complete certainty, the dream of architecture is true.

是啊。你认为质数最神秘的地方是什么？这是个好问题。理论上来说，我们对质数有一个不错的模型。就像我说的，质数有一些明显的模式，比如它们通常是奇数。但除了这些显而易见的规律之外，质数的表现看起来非常随机。假设它们的表现是随机的，有一个叫做质数随机模型的理论。这个模型认为在某个点之后，质数的表现就像一个随机集合。这种模型经过了一些修正，但它非常有效，与数值表现一致，能告诉我们如何预测结果。可以说，完全可以肯定的是，质数的这种“随机”特性确实存在。

The random model gives overwhelming odds that it's true. I just can't prove it. Most of our mathematics is optimized for solving things with patterns in them. And the prime numbers and type pattern as doom holds everything really. But we can't prove that. Yeah, I guess it's not mysterious that the prized wave event is kind of random because there's no reason for them to be to have any kind of secret pattern. But what is mysterious is what is the mechanism that really forces the randomness to happen? And this is just absolute. Another incredibly surprisingly difficult problem is the collots conjecture. Oh yes. Simple to state, beautiful to visualize, in the simplicity, and yet extremely difficult to solve.

随机模型给出了极大的可能性表明这是真的，但我就是无法证明它。我们的大多数数学都是为解决有模式的问题而优化的。然而，质数和类型模式就像末日一样支撑着一切，但我们无法证明这一点。我想，备受重视的波动事件有点随机并不神秘，因为没有理由认为它们会有任何秘密的模式。但是，真正神秘的是到底是什么机制使得这种随机性发生？这一点是绝对的。另一个令人惊讶且极其困难的问题是科拉兹猜想。简单易懂，视觉上也很美，在简单中却极难解决。

And yet you have been able to make progress. Paul Radar said about the collots conjecture that mathematics may not be ready for such problems. Others have stated that it is an extraordinarily difficult problem completely out of reach. This is in 2010 out of reach of present-day mathematics and yet, you have made some progress. Why is it so difficult to make? Can you actually even explain what it is? Oh yeah. So it's a problem that you can explain. It helps with some visual aids. But yeah, so you take any natural number like 13 and you apply the following procedure to it. So if it's even you divide it by two. And if it's odd, you multiply it by three and add one.

尽管如此，你还是取得了一些进展。Paul Radar曾经谈到Collatz猜想时表示，数学可能还未准备好解决这样的问题。其他人也认为这是一个极其困难的问题，完全超出了现有数学的能力范围。这是在2010年，那时现代理论还无法触及这一问题，但是你已经取得了一些进展。为什么这个问题如此难以解决？你能解释一下这是什么问题吗？哦，可以的。这个问题是可以解释的，并且借助一些视觉辅助工具会更好理解。基本上，你可以拿任意一个自然数，比如13，然后对它进行以下操作：如果它是偶数，就把它除以2；如果它是奇数，就乘以3再加1。

So even numbers get smaller, odd numbers get bigger. So 13 would become 40. Because 13 times 3 is 39 and add one of your 40. So it's a simple process for odd numbers and even numbers that both are very easy operations. And then you put together, it's still reasonably simple. But then you ask what happens when you iterate it. You take the output that you just got and feed it back in. So 13 becomes 40. 40 is now even divided by two is 20. 20 is still even divided by 10 to 10. 5 and then 5 times C plus 11 16 and then 8 421. So and then for 1 equals 1421421 is cycles forever.

偶数减小，奇数增大。因此，13会变成40。因为13乘以3等于39，然后加1得到40。对于奇数和偶数来说，这是一个非常简单的过程。把它们组合在一起也依然很简单。但是，当你反复进行这个过程时，会发生什么呢？就是把你刚才得到的输出再输入回去。所以13变成40，40是偶数，除以2得到20，20仍然是偶数，再除以2得到10，5，再加上11得到16，然后是8，4，2，1。然后，1会循环成1，4，2，1，4，2，1，如此反复。

So this sequence I just described 13, 40, 20, 10s or both. These are also called hailstorm sequences because there's an over-stimplified model of hailstorm formation, which is not actually quite correct, but it's some I've taught to high school students as a first-box mission. Is that like a little nugget of ice gets gets a nice crystal forms in cloud and it goes up and down because of the wind and sometimes it's cold it gets a bit more mass and maybe it melts a little bit and this process is going up and down creates this sort of partially melted ice, which I've mentioned because it's health stone. And eventually it falls down to the earth.

这段话描述了一个叫做冰雹序列的过程，其中涉及数字13, 40, 20和10等。这种序列被称为冰雹序列，是因为它与一个过于简化的冰雹形成模型相关，虽然这个模型并不完全正确，但作为一个基础概念，我曾经向高中生教授过。简单来说，这个模型是这样的：一小块冰在云中形成了一个冰晶，然后因为风的影响在上下移动。有时候，它在寒冷的环境中会增加质量，有时候会稍微融化。这个上下移动的过程会产生部分融化的冰块，这就是我提到的冰雹。最终，这块冰雹会降落到地面。

So the conjecture is that no matter how high you start up like you take a number which is in the millions or billions, equal this process that goes up if you're hard and down if you even eventually goes down to to the earth all the time. No matter where you start was very simple algorithmally you end up at one and you might climb for a while. Right. Yeah so it's known. Yeah if you plot it these sequences they look like brownie and motion. They look like the stock market. They just go up and down in a seemingly random pattern.

这个猜想是说，无论你从多大的数字开始，比如以百万或者十亿为单位的数字，根据这样一个过程：如果是奇数就变大，如果是偶数就变小，最终总是会下降到1。不论你从哪里开始，这个简单的算法最终都会让你得到1。虽然在过程中数字可能会上升一段时间。这已经是一个已知的现象。如果你绘制出这些数字序列的图表，它们看起来像布朗运动，或者像股市一样，看似随机地上下波动。

And in fact usually that's what happens. If you plug in a random number you can actually prove that at least initially that it would look like random walk. And actually random walk with a downward drift. It's like if you're always gambling on a net at the casino with odds slightly weighted against you. So sometimes you win sometimes you lose but over in the long run you lose a bit more than you win and so normally your wallet will go to zero if you just keep playing over and over again. So statistically it makes sense.

事实上，通常情况下确实如此。如果你输入一个随机数，你实际上可以证明，至少在初期它看起来就像是随机游走。而且实际上是带有向下趋势的随机游走。这就像是在赌场中赌博，赔率略微对你不利。因此，有时候你会赢，有时候会输，但从长远来看，你输的会比赢的多，所以如果你持续不停地玩，最终你的钱包将会变空。从统计学上来看，这是合理的。

Yes. So the result that I proved roughly speaking asserts that statistically like 19% of all inputs would drift down to maybe not all the way to one but to be much much smaller than what you started. So it's like if I told you that if you go to a casino most of the time you end up if you keep playing up long enough you end up with a smaller amount of any wallet when you start. That's kind of like the result that I proved.

好的。我证明的结果大致来说是这样的：从统计上看，大约19%的输入值会逐渐下降，虽然不一定完全降到1，但会比起始值小很多。这就像是我告诉你，如果你去赌场，大多数情况下，如果你一直玩下去，最终你钱包里的钱会比一开始少。这就是我证明的结果的大概意思。

So why is that result like can you continue down that thread to prove the full conjecture? Well the problem is that I used arguments from probability theory and there's always this exceptional event. So you know so in probability we have this low large numbers which tells you things like if you play a casino with a game at a casino with a losing expectation over time you are guaranteed or almost surely with probability as close to 100% as you wish your guaranteed to lose money.

那么，为什么结果会这样？你能继续沿着这个思路去证明整个猜想吗？问题在于我用了概率论中的论点，而总有一些例外事件。在概率论中，我们有“大数法则”。这个法则告诉我们，比如说，如果你在赌场玩一个预期会输钱的游戏，随着时间的推移，你几乎肯定会输钱，这个概率可以非常接近100%。

But there's always this exceptional outlier like it is mathematically possible that even in the game is also not on your paper you could just keep winning slightly more often than you lose. Very much like how in Navier Stokes there could be you know most of the time your waves can disperse. There could be just one outlier choice of initial conditions that would lead you to and there could be one outlier choice of special number that you stick in that shoes of infinity were all other numbers crashed earth crash to one.

但是，总会有这样的一个特殊例外，就像在数学上是可能的，即使在游戏中，即使你的策略没有写在纸上，你仍有可能略多于输的时候赢。就像在纳维-斯托克斯方程中，大多数情况下你的波动可能会扩散。可能只有一个特殊的初始条件选择会使你走向不同的结果，同样可能有一个特殊的数字选择成为无限的代表，其他数字则趋于崩溃一样。

In fact there's some mathematicians who've Alex Contorovich for instance who've proposed that actually these collects iterations are like these similar automata. Actually if you look at what happened in binary they do if you look a little bit like these game of life type patency and in analogy to how the game of life can create these massive like software pliketing objects and so forth possibly you could create some sort of heavier than air flying machine a number which is actually encoding this machine which is just whose job it is is to encode is to create a version of a software which is larger.

事实上，有些数学家，比如 Alex Contorovich，提出了一个观点：这些科拉兹迭代就像某种类似的自动机。如果研究它们在二进制系统中的表现，会发现它们有点像“生命游戏”这种模式。就像“生命游戏”可以创造出复杂的软件对象一样，这个迭代过程可能也能创造出某些比空气重的飞行器。这个飞行器的本质是通过编码来创建一个更大的软件版本。

Heavier than air machine encoded in a number that flies forever. So Convert in fact work on this problem as well. So Convert so similar in fact that was more on insturations for the Navi Stokes project. That Convert studied generalizations of the collapse problem where instead of more than we're three and adding one or debiting by two you have more complicated branching with us but instead of having two cases maybe you have 17 cases and then you go up and down and he showed that once your iteration gets complicated enough you can actually encode two ing machines and you can actually make these problems understandable and do things like this.

由一个数字编码的重于空气的机器，可以永远飞翔。因此，Convert实际上也在研究这个问题。Convert与Navi Stokes项目的研究非常相似。Convert研究了塌缩问题的推广，在这个问题中，不再是简单的「加一或减去二」，而是有更复杂的分支，也许不只是两种情况，而是可能有17种情况，你可以在其中上下迭代。他展示了一旦迭代过程足够复杂，你实际上可以将其编码成图灵机，并且可以让这些问题变得可以理解，并进行类似的操作。

In fact he met a programming language for these kind of fractional linear transformations. He got a fact track as a play on a full track and he showed that you can program it was two incomplete you could you could you could you could make a program that. If your number you inserted in was encoded as a prime it would sink to zero it would go down otherwise it would go up and things like that. So the general cluster problems is really as complicated as all the mathematics. Some of the mystery of the cellular terminal that we talked about having a mathematical framework to say anything about cellular terminal maybe the same kind of framework is required. Glocks and texture. Yeah if you want to do it not statistically but you really want 100% or all inputs to to to to for the earth.

事实上，他遇到了一种用于处理这种分数线性变换的编程语言。他设计了一种“快捷路径”作为完整路径的变体，并展示了你可以用简化的方式编写程序。如果你插入的数字被编码为一个质数，它将同步到零；否则，它将上升，诸如此类。所以，整体聚类问题真的与所有数学一样复杂。我们之前讨论过的蜂窝终端的一些神秘性，需要一个数学框架来解释它，也许同样的框架是必需的。Glocks和纹理。如果你希望不是通过统计手段，而是希望对所有输入都能达到100%的准确性，那么这种方法是必要的。

Yeah so one might be feasible is yeah assisting 99% you know going to go to one but like everything yeah well that looks hard. What would you say is out of these within reach famous problems is the hardest problem we have today is the remember how about this is we want to it's up there. Pico's MP is a good one because like that's that's a meta problem like if you solve that in the in the positive sense that you can find a Pico's MP algorithm that potentially this solves a lot of other problems as well and we should mention some of the conjectures we've been talking about you know a lot of stuff is built on top of them now.

翻译成中文：是的，有一个可能性是帮助解决 99% 的问题，通常会有一个最优解，但这些往往看起来很困难。你认为在这些著名的问题中，哪个是我们今天面对的最难的问题呢？关于这个问题，我们可能想要解决的是 Pico's NP，这是一个很好的例子，因为这类似于一个元问题。假如你能在这个问题上找到一个有效的算法，就有可能解决许多其他问题。我们还应该提到一些我们一直在讨论的猜想，现在很多事情都是基于这些猜想构建的。

There's a lot of effects. Pico's MP has more ripple effects than basically any other right. If the readman hypothesis is disproven that would be a big mental shock to the number theorists but it would have follow on effects for cryptography because a lot of cryptography uses number theory uses number theory constructions evolving primes and so forth and it relies very much on the intuition that number theorists have built over many many years of what operations evolving primes behave randomly and what ones don't and in particular encryption methods are designed to turn text information on it into text which is indistinguishable from random noise.

这个句子的意思是：影响很大。Pico的MP（某种权利）的连带影响比其他几乎所有权利都要大。如果Readman假说被推翻，对数论专家来说，会是一个巨大的心理冲击，但它也会对密码学产生连带影响。因为许多密码学依赖于数论，数论中的很多构建都涉及到素数等，并且非常依赖数论专家多年来积累的直觉，比如哪些涉及素数的操作是随机表现的，哪些不是。特别是，很多加密方法被设计用来将文本信息变成看起来像随机噪声的文本。

So enhance we believe to be almost impossible to crack at least mathematically but if something has caught our beliefs as we went about this is wrong it means that there are actual patterns at the prime that we're not aware of and if there's one there's probably going more and suddenly a lot of our crypto systems are in doubt but then how do you then say stuff about the primes yeah that you're going towards the collection and conjecture again because if I do you do you want it to be random right yes yeah yeah.

因此，我们认为几乎不可能破解，至少在数学上是这样。但如果我们的这个信念被证明是错误的，这就意味着在质数中存在我们尚未意识到的实际模式。如果发现一个，可能还会有更多，这样一来，我们许多加密系统的安全性就会受到质疑。但是，你如何看待质数呢？如果你尝试收集和猜测模式，那么你希望它们是随机的，对吗？是的，是的。

So more broadly I'm just looking for more tools more ways to show that yeah that things are right no how do you prove a conspiracy doesn't happen is there any chance to you that p equals np is there some can you imagine a possibly universe it is possible i mean this is various scenarios i mean if there's one way it is technically possible but in fact it's never actually implementable the evidence is sort of slightly pushing in favor of no that we probably p is not equit and p i mean it seems like it's one of those cases seem similar to human hypothesis.

更广泛地说，我只是想寻找更多的方法和工具来证明一些事情确实是正确的。你怎么证明某个阴谋没有发生？你认为 P 等于 NP 有可能成立吗？你能想象一个这可能成立的宇宙吗？我的意思是有各种情景，如果有一种方法的话，从技术上来说是可能的，但实际上从未实现过。证据似乎有点倾向于否定，也就是说我们很可能 P 不等于 NP。这似乎像是一种与人类假设类似的情况。

I think the evidence is leading pretty heavily on the know certainly more on the know than on the yes the funny thing about p equals np is that we have also a lot more obstructions than we do for almost any other problem so while there's evidence we also have a lot of results ruling out many many types of approaches to the problem. This is the one thing that the computer science is actually very good at it's actually saying that certain approaches cannot no go theorems it could be understandable we don't we don't know.

我认为证据更倾向于“否定”，至少比“肯定答案”多。关于“P等于NP”的有趣之处在于，相较于其他问题，我们在这个问题上更多地遇到了各种障碍。因此，虽然有一些证据，但我们也有很多结果表明许多类型的方法都是行不通的。在这方面，计算机科学确实很擅长，擅长于证明某些方法不可行。不过，有些结论可能我们仍然不清楚。

There's a funny story I read that when you won the fields metal somebody from the internet wrote you and asked uh you know what are you going to do now they want this procedure award and then you just quickly very humbly said that you know this shiny metal is not going to solve any of the problems i'll currently working out some okay keep working on them it's just first of all it's funny to me that you would answer an email in that context and second of all it um it just shows your humility but anyway uh maybe you could speak to the fields metal but it's another way for me to ask about uh about Gregor at problem and what do you think about him famously declining the fields metal and the millennial prize which came with a one million dollar of prize money he stated that i'm not interested in money or fame the prize is completely irrelevant for me if the proof is correct then no other recognition is needed.

我读过一个有趣的故事，说当你获得菲尔兹奖之后，网上有人给你写信，问你现在打算怎么做，因为你赢得了这个有名的奖项。你简单而谦逊地回复说，这块闪亮的奖牌不会解决我现在正在研究的问题，所以还是继续努力工作。这让我觉得有趣的是，你会在这样的情况下回复邮件，其次，这也显示了你的谦逊。不过，也许你可以谈谈菲尔兹奖，这也是我转而询问你对格里戈里·佩雷尔曼的问题的看法。他因拒绝菲尔兹奖和附带百万美元奖金的千禧年大奖而闻名。他表示：“我对金钱和名声没有兴趣。对我来说，奖项是完全无关紧要的。如果证明是正确的，就不需要其他认可。”

Yeah well he's he's somewhat of an outlier um even among mathematicians who tend to uh do have uh someone idealistic views um i've never met him i think i'd be interested in me to one day but i'd never have the chance i know people who met him he's always had strong views on certain things um you know i mean it's not like he was completely isolated from the math community i mean he would give talks and papers and so forth but somewhere he just decided not to engage with the rest like i mean he was dissolution of something um i don't know um and he decided to uh to piece out uh and you know collect mushrooms and same papers work or something and that's that's fine you know and you can you can do that um i mean that's another sort of flip side i mean we are not a lot of problems that we solve you know they some of them do have practical application that's that's great but uh like if you stop thinking about a problem you know so he's he hasn't published since in this field but that's fine there's many many other people you've done as well um.

嗯，他在某种程度上是个异类，甚至在倾向于拥有某种理想主义观点的数学家中也是。我没见过他，不过我想有一天能见到他会很有趣，但可能没有机会。我认识一些见过他的人，他们说他在某些事情上始终有很强烈的观点。虽然他并没有完全与数学界隔绝，比如他会发表演讲和论文，但在某个时刻，他似乎决定不再与别人交流，可能是对某些事情失望了吧。我不太清楚。他决定选择另一种生活，例如采蘑菇，继续写论文或做其他什么事情。我觉得这样挺好的，你可以这么做。我们在解决问题中并不是所有问题都有实际应用，当然有些有，那就很好。但如果你不再考虑某个问题也是可以的。他后来在这个领域没有再发表过什么，不过这没关系，还有许多其他人继续在做。

Yeah so I guess one thing I didn't realize initially with the field's metal is that it sort of makes you part of the establishment um you know so you know most mathematicians you have this it's just career mathematicians you know you just focus on publishing your next paper maybe getting one just to promote a one one rank you know and and starting a few projects maybe having taking some students or something yeah but then suddenly people want your opinion on things and you have to think a little bit about uh you know things that you might just have foolishly say because you know no one's gonna listen to you uh this is so it's more important now is it constraining to you are you able to still have fun and be a rebel and try crazy stuff and plot uh play with ideas.

好的，所以我想我最初没有意识到的是，获得菲尔兹奖会让你成为学术权威的一部分。大多数数学家通常只是职业数学家，他们专注于发表下一篇论文，也许升职一级，开启一些项目，也许带一些学生。但是，突然之间，人们会想听取你对各种事情的看法，这时你就得多考虑一下，不再能随意发表一些可能不太成熟的言论，因为现在有人会听你的。这是不是对你有束缚呢？你还能否继续享受乐趣，做个"叛逆者"，尝试一些疯狂的想法，并且玩味这些创意？

I have a lot less free time than I had previously um I mean mostly by choice I mean I I I I can always see I have the option to sort of decline so I declare a lot of things I could decline even more um well I could acquire a repetition things so unreliable that you would have even asked anymore uh this is this I love the different algorithms here this is great this is it's always an option um but you know um there are things that are like I mean so I mean I I don't spend as much time as I do as a postdoc you know just working the one part of the time or um falling around I still do that a little bit but yeah as you're advancing your career it's some of the more soft skills so math somehow front nodes all the technical skills to the early stages of your career so um yeah so it's uh as a postdoc is publisher parish you're you're you're incentivized to basically focus on on proving very technical themes to prove yourself um as well as prove the theorems um.

我现在的空闲时间比以前少了很多，主要是因为自己的选择。我可以选择拒绝一些事情，实际上我确实拒绝了很多事情，不过我还可以更多地拒绝。有时候，我会犹豫不决，但这个犹豫本身就是一种选择。现在，我没有像做博士后那样大部分时间都在专注于一个项目或者随意地消磨时间。虽然我还是偶尔这样做，但职业生涯的进步让我开始重视一些软技能。在职业生涯的早期阶段，数学和技术技能会更重要。所以在博士后阶段，学术界有“发表或淘汰”的压力，大家都被激励去证明自己，发表技术性强的成果，同时也证明各种数学定理。

But then as as you get more senior you have to start you know mentoring and and and giving interviews and uh and trying to shape direction of field both research wise and and you know uh sometimes you have to uh uh you know to present with administrative things and it's kind of the the right social contract because you you need to to work in the trenches to see what can help mathematicians the other side of the establishment sort of the the really positive thing is that um you get to be a light that's an inspiration to a lot of young mathematicians and young people that are just interesting mathematics and it's like yeah yeah it's just how the human mind works this is where i would probably uh say that i like to field metal that it does inspire a lot of young people somehow i don't this is just how human brains work.

翻译成中文：但是，当你变得更资深时，你就需要开始进行指导、面试，并尝试在研究方面以及其他方面为领域设定方向。有时你还需要处理一些行政事务。这是一种合理的社会契约，因为你需要在基层工作，以发现什么能够真正帮助数学家。另一方面，积极的一面是，你能够成为启发很多年轻数学家和对数学感兴趣的年轻人的光辉榜样。这可能也是为什么我欣赏菲尔兹奖，因为它的确能够激励许多年轻人，这就是人类思维运作的方式。

Yeah at the same time I also want to give sort of respect to somebody like Gore Promen who is critical of awards in his mind those are his principles and any human that's able for their principles to like do the thing their most humans would not be able to do it's beautiful to see some recognition is it's necessary and important uh but yeah it's it's also important to not let these things take over your life um and like only be concerned about getting the next big award or whatever um I mean yeah yeah yeah so again you see these people try to only solve like a really big math problems and not work on on on things that are less uh sexy you wish but but but actually uh it's still interesting and it's instructive as you say like the way the human mind works it's um we understand things better when they're attached to humans um and also uh if they're attached to a small number of humans like so I mean the the the the way our humans um minus is is why we can comprehend the relationship between the 10 or 20 people.

同时，我也想表达对像戈尔·普罗门这样的人的尊重，他对奖项持批评态度，那是他的原则。任何能坚持自己原则的人，即使是去做大多数人无法做到的事情，都是值得赞美的。虽然得到一些认可是必要和重要的，但是不应该让这些东西主导你的生活，不应该只关注于获得下一个大奖。很多人试图只解决一些非常重大的数学问题，而忽视那些看似不那么吸引人的事情，但其实这些问题也很有趣并且富有启发性。正如你所说，人类的大脑理解事物的方式是通过联结到某个人或者少数几个人来实现的。这样，我们就能更好地认识和理解人与人之间的关系。

But once you get beyond like a hundred people like this is this is a there's a limit I figured this name for it um beyond which uh it just becomes the other um and uh so we have you have to simplify the poem master you know nine point nervous energy becomes the other um and uh and of these models are incorrect in this causes or kinds of problems. but um so yeah so to humanize a subject you know if you identify a small number of people i say you know these are representative people of the subject you i say role models for example um that has some role um but it can also be um uh yeah too much of it can be harmful because it's i'll be the first to say that my own career trough this not that of a typical mathematician um i the very accelerated education i skipped a lot of classes.

一旦你面对的人数超过一百，就会有一个极限，我给它起了个名字，超出这个范围后，事情就会变得模糊，变成“其他”。因此，我们需要简化，诗意地说，九点几的紧张能量就变成了“其他”。这些模型不准确，会引发各种问题。所以，为了让一个主题更具人性化，你可以挑选出一些人，称他们为这个主题的代表性人物，或者说榜样。这确实有一定的作用，但过度也可能带来不良影响。我一定会先说，我自己的职业生涯并不是一个典型数学家的发展路线，因为我的教育历程非常加速，跳过了很多课程。

Um i think i was very fortunate mentoring opportunities um and i think i was at the right place at the right time just because someone doesn't have my um to lectury you know it doesn't mean that they can't be good mathematicians i mean they may be better just in a very different style uh and we need people with different style um and you know even and sometimes too much focus is given on the on the on the person who that's the last step to complete um project in mathematics or elsewhere that's that's really taken you know centuries or decades with lots and of putting a lot of previous work um but that's a story that's difficult to tell um if you're not expert because you know it's easy to just say one person did this one thing you know it makes more much simpler history.

嗯，我觉得我很幸运，得到了很多指导的机会，我当时也在合适的时间出现在了合适的地方。因为有人可能没有我的讲课方式，但这并不意味着他们不能成为优秀的数学家。他们可能以完全不同的风格表现得更好，而我们确实需要有不同风格的人。有时候，人们过于关注那些完成数学或其他领域项目最后一步的人。但实际上，这些项目可能耗费了数十年甚至几个世纪，并且需要大量前人的积累和努力。如果你不是专家，很难讲述这样的故事，因为单说某个人做了某件事会使历史显得简单得多。

I think on the whole it um is a hugely positive thing to to talk about Steve Jobs as a representative of apple when i personally know and of course i've already knows the incredible design the incredible engineering teams just the individual humans on those teams they're not a team they're individual humans on a team and there's a lot of brilliance there but it's just a nice shorthand like a very like pie yeah Steve Jobs yeah yeah pie as a starting point you see yeah as a first approximation that's how you read some biographies and then look into much deeper first approximation yeah that's right uh so you must you were a person to um angel wiles at that time oh yeah professor there.

我认为，总的来说，把史蒂夫·乔布斯作为苹果公司的代表来谈论是一件非常积极的事情。当我个人了解、并且当然已经知道苹果那令人惊叹的设计和工程团队——这些团队中的每一个个体都是出色的，虽然他们组成了一个团队，但更重要的是，他们每个人都是独特的个体，才华横溢。然而，把史蒂夫·乔布斯作为一个象征性代表是很方便的，就像一个简化的模型，比如说“苹果派”那样，你懂的。作为一个起点来看待问题，是一种很好的方法，就像读一些传记，然后再深入了解。没错，当时你是Angel Wiles的人，是在那儿的教授对吧？

It's a funny moment how history is just all interconnected and at that time he announced that he proved the from us last year what did you think maybe looking back now with more context about that moment in math history yeah so i was a graduate student at the time i mean i i vaguely remember you know there was press attention and uh um we all had the same um we heard pigeonholes in the same mail group you know so we all particularly on mail and like suddenly angel wiles is mailbox exploded to be all the flowing that's a good metric yeah um you know so yeah we all talked about it at t and so forth i mean we didn't understand most of us sort of understand the proof um we understand sort of high level details.

这是一个有趣的时刻，展示了历史的相互关联性。在那个时候，有人宣布证明了我们去年提出的某个问题。回顾当时的数学历史，现在更具背景知识的你怎么看待那一刻？我当时还是个研究生，我隐约记得有媒体的关注。我们都在同一个邮件组里，所以我们特别关注邮件。突然之间，安德鲁·怀尔斯的信箱里塞满了各种信息，这是个好兆头。我们都在讨论这件事，同时也在其他地方交流。其实大多数人并不完全理解这个证明，只是了解了一些高层次的细节。

Um like there's an ongoing project to formalize it in lean like Kevin Bonson is actually yeah can we take that small tangent is it is it how difficult does that because as i understand the from us that the proof for from as last term has like super complicated objects yeah yeah really difficult to formalize now yeah i guess yeah you're right the objects that they use um you can define them uh so they've been defined in lean okay so so just defining what they are can be done uh that's really not trivial but as we've done there but there's a lot of really basic facts about um these objects that have taken decades to prove in that they're in all these different math papers and so a lot of lots of these have reformalize as well.

嗯，其实有一个正在进行的项目，目的是用 Lean 形式化这个内容，比如 Kevin Bonson 就参与其中。我们可以稍微谈谈这个吗？就是我想知道这有多难，因为据我所知，那些证明里包含了一些非常复杂的对象，是吧，是啊，现在确实很难形式化。我想是的，你说得对，他们使用的这些对象是可以在 Lean 里面定义出来的，所以定义它们是可以做到的，这并不简单，但我们已经完成这一步了。不过，这些对象有很多非常基础的事实，它们在很多不同的数学论文中经过了几十年的证明。所以，我们也需要重新形式化这些内容。

Um Kevin's uh Kevin Bonson's goal actually he has five year grad to formalize philosophy and his aim is that he doesn't think he could be able to get all the way down to the basic axioms but he wants to formalize it to the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to um to number theories at the time um and then some other person or some other work we're happy done to to to get from there um so it's it's a different area mathematics than um the type of mathematics i'm used to um in analysis which is kind of my area um the objects we study are kind of much closer to the ground we study I study things like prime numbers and and and functions and and things that i wouldn't scope of a high school math education to at least define um yeah but then this is very advanced algebraic side of number theory where people have been building structures of more structures for quite a while and it's a very sturdy structure this it's it's been it's been very um at the base it releases extremely what about the text books and so forth but um it does get to the point where if you if you haven't taken these years of study and you want to ask about what what is going on at like level six of the office tower.

Kevin Bonson的目标是，用五年的时间将哲学形式化。他的意图是，即使无法彻底简化到最基本的公理，他也希望能够将其形式化到一个程度，即他所需依赖的“黑箱”知识仅限于1980年前数论中已知的内容以及其他某些人的研究成果。这与我熟悉的数学领域不同，我的研究方向是分析学。我们研究的对象更接近基础，比如质数、函数等，这是高中数学教育范围内至少能够定义的内容。然而，Kevin所研究的是非常高级的代数数论，那里的人们已经构建了非常坚固的结构，基础知识经过长时间的发展和巩固。如果没有经过多年的学习，很难理解像“办公塔楼第六层”这样的高级概念是什么。

I yeah spend quite a bit of time before they can even get to the point where you can see you see some of you recognize water inspires you about his journey that was similar as we talked about seven years most working in secret yeah uh yes that is a eromatic yeah so it kind of fits with sort of the the romantic image that people have of mathematicians to be extent they think of things that are at all as these kind of eccentric you know wizards or something um so that certainly kind of a uh accentuated that perspective you know I mean it's it is a great achievement a hairstyle of solving problems is so different from my own um but which is great I mean we we need people like that speaker like what uh in in terms of like the you like the collaborative I like moving on from a problem if it's giving too much difficulty um but you need the people who have the tenacity and the the fearlessness um you know I’ve collaborated with with people like that where well I want to give up uh because the first approach that we tried didn't work in the second one didn't approach they convinced and they have a third fourth and the fifth of what works um and I'd have to eat my work okay I didn't think this was going to work but yes you will write along and we should say for people don't know not only are you known for the brilliance of your work but the incredible productivity just the number of papers which are all of very high quality so there's something to be said about being able to jump from topic to topic.

我花了相当多的时间才意识到，有些你认识的人水到渠成，他们的旅程中有些东西能激励你，我们提到过七年时间大多是在秘密中工作的。是的，这与人们对数学家那种浪漫的形象不谋而合，他们被认为是有些古怪的"巫师"。这无疑加深了这种印象。这真是一个伟大的成就，解决问题的方式和我自身的风格截然不同，但这也很好。我们确实需要这样的人，他们会持续探索。而我通常会选择放弃那些实在太过困难的问题。但我们需要那些有毅力和无所畏惧的人。我曾经和这样的人合作过，当我想放弃的时候，他们却不断尝试各种办法，直到第三、第四甚至第五种办法奏效时，我不得不承认自己错了。这些人通常会坚持到最后，并且他们通常是对的。另外，对于那些不太了解您的人，您不仅以工作的卓越而闻名，同时也是因为您惊人的多产，每篇论文都具有高质量。这种能够跨越不同主题的能力真的很值得称赞。

Yeah it works for me yeah I mean I'm being also people who are very productive and they take focus very deeply on yeah I think I want us to find their own workflow um like one thing which is a shame in mathematics is that we have mathematics there's sort of a one-size-fits-all approach to teaching mathematics um and you know so we have a certain curriculum and so forth I mean you know maybe like if you do math competitions or something you get a slightly different experience but um I think many people um they don't find their their native math language uh until very late or usually too late so they they they stop doing mathematics and they have a bad experience with a teacher who's trying to teach the one way to do mathematics that they don't like it um

是的，这对我来说可行。我指的是，我也遇到过一些非常高效，且专注于他们工作的人。是的，我认为我们应该找到自己的工作流程。其中一个关于数学的遗憾是，我们在数学教学上采取了一种“一刀切”的方法。我们有一个固定的课程等等。虽然参加数学竞赛可能会有一些不同的体验，但我认为很多人直到很晚甚至通常太晚才找到他们自己真正擅长的数学语言。因此，他们停止学习数学，因为在学校期间，某位老师试图用一种他们不喜欢的方式教授数学，这给他们带来了不好的经历。

My theory is that um humans don't come evolution has not given us a math center or a brain directly we have a vision center and a language center and some other centers um which have evolution as home but we it doesn't we don't have in eight sense of mathematics um but our other centers are sophisticated enough that different people uh we we can repurpose other areas or a brain to do mathematics so some people have figured out how to use the visual. center to do mathematics and so they think very visually when they do mathematics some people have to repurpose their their language center and they think very symbolically um

我的理论是，人类在进化过程中并没有获得一个专门的数学中心或大脑直接处理数学的部分。我们有视觉中心和语言中心以及其他由进化决定的部分，但我们并没有与生俱来的数学感知能力。然而，我们的大脑其他部分足够复杂，以至于不同的人可以重新利用大脑的其他区域来处理数学问题。有些人学会了如何利用视觉中心来进行数学思考，因此他们在进行数学问题时会以视觉的方式来思考。有些人则需要重新利用他们的语言中心，因此他们在数学思维中采用更多符号化的方式。

You know um some people like if they are very competitive and they feel like gaming they there's a tap there's a part of the brain that's very good at at at uh at solving puzzles and games and and and that can be repurposed but like when I talked about the mathematicians you know they don't quite think they i can tell that they're using some of different styles of thinking then I i mean not not destroyed but they they may prefer visual like i i i don't actually prefer visual so much i need also visual aids myself um you know mathematics provides the common language so we can still talk to each other even if we are thinking in different ways but you can tell there's a different set of subsystems being used in the thinking process like they take it from past they're very quick at things that i struggle with and vice versa um and yet they still get to the same goal um that's beautiful and

你知道，有些人非常有竞争性，当他们玩游戏时，大脑中负责解决谜题和游戏的部分会被激发，并且这些能力可以被重新利用。但是，当我谈到数学家时，我能感觉到他们在思考时使用的是不同的思维方式。他们可能更喜欢使用视觉辅助，而我自己其实并不太偏好视觉，但我也需要一些视觉帮助。数学提供了一种通用的语言，让我们即使以不同的方式思考也能互相交流。不过，你会发现他们在思考过程中使用了一组不同的子系统，比如，他们对于我觉得困难的事情能很快解决，反之亦然，但我们最终都能达到相同的目标。这样的过程很美妙，而且。

Yeah but i mean the way we educate unless you have like a person like two to or something i mean education sort of just by nature of scale has to be mass produced you know you have to teach to 30 kids you know they have 30 different styles you can't you can't teach 30 different ways on that topic what advice would you give to students uh young students who are struggling with math and but are interested in it and would like to get better is there something in this yeah in this complicated educational context what would you

是啊，我的意思是，我们的教育方式，除非你有一个像导师那样的存在，否则教育这种东西由于规模的原因，往往是要大规模进行的。你知道的，你得教30个孩子，他们有30种不同的学习方式，你没办法就一个主题用30种不同的方法来教。对于那些在数学上有困难但对其感兴趣并希望提高的年轻学生，你有什么建议吗？在这种复杂的教育背景下，你会怎么做呢？

Yeah it's a tricky problem one nice thing is that there are now lots of sources for my faculty in Richmond outside the classroom um so in my day they're already there are math competitions um and you know they're also like popular math books in the library um yeah but now you have you know youtube uh there are forums just devoted to solving you know math puzzles and um and math shows up in another place you know like um for example there are hobbyists who play poker for fun and um they they you know they are for very specific reasons are interested in very specific probability questions um yes and and they they actually know there's a community of amateur probabilists in in poker um in chess and baseball i mean there's there's uh yeah um there's math all over the place um

是的，这是个棘手的问题。不过有一件好事，就是现在在里士满，我的同事们在课堂外有很多资源。在我那个年代，已经有数学竞赛了，图书馆里也有受欢迎的数学书籍。而现在，你有了YouTube，还有专门致力于解决数学谜题的论坛。数学还出现在其他地方，比如有些爱好者为了娱乐玩扑克，他们对特定的概率问题很感兴趣，因此扑克领域有一群业余概率学者。在国际象棋和棒球中也是如此，可以说数学无处不在。

And i'm hoping actually with there with these new sort of tools for lean and so forth that actually we can incorporate the broader public into math research projects um like this is almost is doesn't happen at all currently so in the sciences there's some scope for citizen science like astronomers uh they're amateurs who would discover comets and there's biologists there are people who could identify butterflies and so forth um and in math there are smaller activities where um amateur mathematicians can like discover new primes and so forth but but previously because we have to verify every single contribution um like most mathematical research projects it would not help to have input from the general public in fact it would it would just be be time consuming because just error checking and everything um

我希望通过这些新的精益工具等方式，我们能够把更广泛的公众纳入到数学研究项目中去。目前这种情况几乎不存在。在科学领域，一些领域有业余科学家的参与，比如天文学中，业余爱好者可以发现彗星，生物学中，人们可以识别蝴蝶等等。在数学领域，也有一些小规模的活动，比如业余数学爱好者可以发现新的素数等。但以前，由于我们必须验证每一个贡献，大多数数学研究项目并不适合公众的参与。实际上，这样做反而会很耗时，因为需要进行错误检查等工作。

But you know one thing about these formalization projects is that they are bringing together more bringing in more people so i'm sure there are high school students who have already contributed to some of these formalizing projects who contributed to math with um you know you don't need to be a PhD holder to just work on one atomic thing there's something about the formalization here that also it's it's a very first step opens it up to the programming community yes the people who are already comfortable with program it seems like programming is somehow maybe just the feeling but it feels more accessible to folks than math math is seen as this like extreme especially modern mathematics seen as this extremely difficult to enter area and programming is not so that could be just an entry point you can actually code and it can get results you know you can put out the world pretty quickly.

这些形式化项目有一个好处，就是它们吸引了越来越多的人参与进来。我相信已经有高中生为其中一些形式化项目做出贡献，他们在数学方面也有所贡献。你知道的，不需要有博士学位就能在某个小问题上工作。这里的形式化有一点非常重要，它为编程社区打开了大门。那些已经对编程感到舒适的人，可能会觉得编程比数学更容易接近。数学，尤其是现代数学，常常被视为一个极难进入的领域，而编程似乎不是这样。因此，这可能是一个进入的切入点。通过编写代码，你实际上可以获得成果，并可以很快将其应用于现实世界。

Yeah, um, yeah like if uh if programming was taught as an almost entirely theoretical subject where you just talk with the computer science the theory of functions and and and and routines and so forth and and outside of some some very specialized homework assignments you don't like your program like on the weekend for fun yeah or yeah there would be as considered as hardest math.

是的，嗯，是的，就像如果编程被教得几乎完全是理论性的科目，只涉及计算机科学中的函数理论、例程等等，而且除了少数一些非常专业的作业之外，你不会在周末为了乐趣去编写程序。那么，它会被视为像高难度的数学一样困难。

Um yeah so as I said you know there are communities of non mathematicians where they're deploying math for some very specific purpose you know like like optimizing the poker game and and for them then math becomes fun for them.

嗯，是的，就像我之前说的，有一些非数学专业的群体，他们为了某些非常具体的目的使用数学，比如优化扑克游戏。对于他们来说，数学因此变得有趣起来。

What advice would you give in general to young people how to pick a career how to find themselves like yeah that's a tough tough tough question yeah so um there's a lot of certainty now in the world you know I mean I I there was this period after the war where uh at least in the west you know if you came from a good demographic you uh you know like you there was very stable path through it to a good creator you go to college you get an education you pick one profession and you stick to it it's becoming much more think of the past.

对于年轻人来说，如何选择职业和找到自我是一件很困难的事情。现在的世界有很多不确定性。在战争结束后的那段时间，至少在西方国家，如果你有好的背景，通常会有一条很稳定的道路通向成功：你上大学，接受教育，选择一个职业，并长期坚持下去。但这种情况正在变得越来越像过去的事情。

So I think you just have to be adaptable and flexible I think people will have to get skills that are transferable you know like learning one specific program in language or one specific subject mathematics or something it's that is still this not a super transferable skill but sort of knowing how to um the reason with abstract concepts or how to problem solve and things go wrong or so anyway these are things which I think we will still need even as our tools get better.

所以，我认为你必须具备适应性和灵活性。我认为人们需要获得可转移的技能，比如学习一种特定的编程语言或某个具体的学科，比如数学，这些虽然重要，但并不是非常具备可转移性的技能。相反，理解抽象概念、解决问题的能力在事情出错时尤为重要。我认为即使我们的工具在不断进步，这些能力仍然是我们需要的。

And you know you you'll be working with AI sports so forth but actually you're an interesting case study I mean you're like one of the great living mathematicians right and then you had a way of doing things and then all of a sudden you start learning I mean first of all you kept learning new fields but you've learned lean that's not that's a non trivial thing to learn like that's uh yeah that's a for a lot of people that's extremely uncomfortable leap to take right yeah so mathematicians um let's well I've always been interested in new ways to do mathematics.

你知道你会与人工智能和体育等领域一起工作，但实际上，你是一个有趣的案例。我是说，你是当代伟大的数学家之一，对吧，然后你有自己的一套做事方式。突然之间，你开始不断学习新领域。首先，你一直在学习新的领域。而且，你还学会了Lean，这可不是一件简单的事情。对很多人来说，这是一个非常艰难的跨越，对吧。数学家...嗯，我一直对探索新的数学方法感兴趣。

I I feel like a lot of the ways we do things right now are inefficient um like I spend like many of my colleagues you spend a lot of time doing very routine computations or doing things that other mathematicians would instantly know how to do and we don't know how to do and why can't we search and get a quick response and so on so that's why I've always been interested in exploring new workflows about four or five years ago I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute and at the time Peter Schultzer had just formalized one of his new theorems and there's some other developments in computer assisted proof that look quite interesting and I said oh we should we should uh we should have a workshop on this this is a good idea.

我觉得我们目前做事情的很多方法都很低效，就像我和我的许多同事一样，你会花很多时间在一些非常常规的计算上，或者是在做一些其他数学家会立刻知道怎么做的事情，而我们却不知道。为什么我们不能通过搜索快速得到答案呢？所以我一直对探索新的工作流程很感兴趣。大约四五年前，我在一个委员会中，需要为数学研究所策划有趣的研讨会。当时，彼得·舒尔策刚刚形式化了他的一个新定理，还有一些计算机辅助证明的发展看起来相当有趣。我说，我们应该举办一个关于这方面的研讨会，这是个好主意。

Um and then I was a bit too enthusiastic about this idea so I got violent told to actually write it um so I did with a bunch of other people Kevin Pozzett and Jordan Ellen Bergen and uh on a bunch of other people um and it was it wasn't a nice success we brought together a bunch of mathematicians and computer scientists and other people and and we got up spin on state VR um and it was really interesting um developments that but most mathematicians didn't know what was going on um that lost a nice proofs of concept you know it's just so hints of what was going to happen.

嗯，然后我对这个想法有点过于热情，所以有人强烈建议我实际去写下来。于是，我和其他一些人，包括Kevin Pozzett和Jordan Ellen Bergen一起动手了。我们汇集了一群数学家、计算机科学家和其他人，最终在VR状态方面取得了一些进展。这是一次非常有趣的发展，但大多数数学家并不清楚具体发生了什么。虽然我们获得了一些不错的概念验证，但这只是对未来可能发生事情的一些初步提示。

This was just before chat GVT but there was even then there was one talk about language models and the potential uh of those in the future so that got me excited about the subject so I started giving talks um about this is some of which more of us just started looking at um now that I mentioned the run as conference and then chat GVT came out and like suddenly AI was everywhere and so uh I got interviewed a lot um about about this topic um and in particular um the interaction between AI and formal proof systems.

这件事情发生在Chat GVT之前，但即使在那时，也已经有关于语言模型及其未来潜力的讨论。这引起了我对这个主题的兴趣，所以我开始做一些相关的演讲。其中一些是我们才开始关注的，现在提到会议运行时，然后Chat GVT出来了，突然之间AI无处不在。因此，我在这个主题上接受了很多采访，特别是关于AI和形式化证明系统之间的互动。

And I said yeah they should be combined this this this this is um this perfect synergy to happen here and at some point I realized that I have to actually do not just talk the talk but walk the walk you know like you know I don't work in machine learning I don't work in through formalization and there's a limit to how much I can just rely on authority and saying you know I'm a I'm a warnaut mathematician just trust me you know when I say that this is going to change mathematics and I'm not doing it any and I don't do any of it myself.

我说，没错，它们应该结合在一起，这将会产生完美的协同效应。后来我意识到，我不能光说不做，而是要身体力行。你知道，我不是从事机器学习工作，也不是搞形式化的工作，我在这一领域能依赖的权威有限。不能只是说“我是个数学家，请相信我”，就声称这将改变数学，然而我自己却没有参与其中。

So I thought like I had to actually uh uh justify it yeah but there's a lot of what I get into actually um I don't quite see an advice as how much time I'm gonna spend on it and it's only after I'm sort of waste deep in in in a project that I I realized by that point I'm committed well that's deeply admirable that you're willing to go into the fray be in some small way beginner right or have some of the sort of challenges that a beginner would right and yeah new concepts new ways of thinking also you know sucking at a thing that others.

所以我当时觉得我必须要为此辩解，但其实我投入的很多事情，我没有真正看到一个清晰的建议说明我需要花多少时间在上面。只有当我已经深入到项目中时，我才意识到到那时我已经投入其中了。你愿意勇敢去面对那些挑战，哪怕有些地方像个新手，这真是令人钦佩。是的，新的概念，新的思维方式，还包括做一些别人可能更擅长的事情时遇到的困难。

I think I think in that talk you know you could be a feels no matter winning mathematician and undergrad knows something better yeah um I think mathematics inherently I mean mathematics is so huge these days that nobody knows all of modern mathematics and inevitably we make mistakes and um you know you can't cover up your mistakes with just sort of provide or and I mean because people will ask for your proofs and if you don't have the proofs you know what the proofs um I don't love math.

我觉得，在那次谈话中，我意识到即使你是一个获奖的数学家或者是个本科生，也可能会犯错。数学本身是非常庞大的领域，现代数学发展的如此迅速，以至于没有人能掌握所有内容，因此我们很难避免犯错。而且，你不能仅靠说辞掩盖错误，因为别人会要求你提供证明，如果你没有证明，事情就会暴露。我不热爱数学。

Yeah so it does keep us honest I mean not I mean you can still it's not a perfect uh panacea but I think uh we do have more of a culture of admitting error then because we're forced to all the time big ridiculous question I'm sorry for it once again who is the greatest mathematician of all time maybe one who's no longer with us who are the candidates the oiler gouse newton ramanogen helper.

好的，这确实让我们保持诚实，我的意思是，这并不是一个完美的解决方案，但我认为我们确实拥有更多愿意承认错误的文化，因为我们总是被迫这样做。抱歉，接下来是一个很难回答的问题：有史以来最伟大的数学家是谁？可能是一位已经不在世上的数学家。候选人可能有欧拉、高斯、牛顿、拉马努金等人。

So first of all I say I said much before like there's some time dependence but on the day yeah like if you if you if you park cumulatively over time for example you click like like like so like this is one of the good tenders um and then maybe some unnamed anonymous method just before that um you know whoever came up with the concept of obnumbus you know um do mathematicians today still feel the impact of Hilbert just oh yeah directly of everything that's happened in the 20th century.

首先，我之前已经多次提到过，某些事情是有时间依赖性的。比如说，如果你在一段时间内累计停车，例如你点了很多次"点赞"，这就是一个好的策略。然后，在此之前可能有一些未命名的匿名方法。不管是谁提出了"obnumbus"这个概念，今天的数学家们是否仍然感受到希尔伯特的影响？哦，是的，希尔伯特对二十世纪发生的一切都有着直接的影响。

Yeah Hilbert spaces we have lots of things that are named after him of course just the arrangement of mathematics and just the introduction of certain concepts I mean 23 problems have been extremely influential there's some strange power to the declaring which problems yeah hard to solve the statement of the open problems yeah I mean you know this is bystander effect everywhere like if no one says you should do x and I just sort of mills around. winning for someone else to to do something and like nothing gets done.

是的，关于希尔伯特空间，我们有很多东西是以他命名的。当然，这只是数学的安排和一些概念的引入。希尔伯特提出的23个问题影响深远。确实在于他指出了哪些问题难以解决，这其中有某种奇妙的力量。提出未解决的问题的声明是很重要的。比如说，这就像旁观者效应。如果没有人说你应该去做某件事，我可能就会不断地徘徊，等待别人去做，结果什么都没有完成。

Um so and and like it's the one one thing that actually uh you have to teach undergraduates in mathematics is that you should always try something so um you see a lot of paralysis in an undergraduate trying a math problem if they recognize that there's a certain technique that can be applied they will try it but there are problems for which they see none of their standard techniques obviously applies and the common reaction is then just paralysis I don't know what to do.

在数学教学中，有一件事情是必须教给大学本科生的，那就是他们应该始终尝试去做某事。你会发现，当本科学习生遇到数学问题时，如果他们知道有某种技巧可以应用，他们会去尝试。但是，有些问题看起来没有任何标准技巧可以明显应用，学生的常见反应是无从下手的瘫痪状态。我不知道该怎么办。

Oh um I think there's a quote from the Simpson so I've tried nothing and I'm all that of ideas so you know like the next step then is to try anything like no matter how stupid um and in fact how almost the stupid are the better um which you know I want a technique which is almost guaranteed to fail but the way it fails is going to be instructive um like it fails because you do not at all taking to account this hypothesis oh this hypothesis must be useful that's a clue.

哦，嗯，我记得《辛普森一家》有一句台词说，我什么都没试过，现在已经没有主意了。所以，接下来就是什么都试试看，不管多愚蠢的点子。实际上，越愚蠢越好。我想要一种几乎肯定会失败的方法，但失败的方式能给我启发。比如，因为完全没有考虑某个假设而失败，这就表明那个假设可能是有用的，这是个提示。

I think you also suggested somewhere this this fascinating approach which really stuck with me as they're using it it really works I think you said it's called structured procrastination no yes it's when you really don't want to do a thing the imagine a thing you don't want to do more yes yes it's worse than that and then in that way you procrastinate by not doing the thing that's worse yeah yeah it's a nice it's a nice hack it actually works yeah.

我记得你曾经提到过一种很有趣的方法，这个想法让我印象深刻，因为他们使用这种方法真的很有效。我记得你说它叫做“结构性拖延症”，对吗？就是当你真的不想做某件事时，就想象出一个你更不想做的事情，是的，那事情更糟糕。这样一来，你就通过不做那件更糟糕的事情来拖延。这是一个很不错的小技巧，实际很有效。

There's um I mean with anything like you know I mean like you um psychology is really important by you you you talk to athletes like marathon runners and so forth and yeah and they talk about what's the most important thing is that they're training veteran men or the diet and so on so much of it is like psychology um you know just tricking yourself to to think the the most feasible um so you you're motivated to do it.

心理学在很多事情中都很重要，比如当你和马拉松选手这样的运动员交流时，他们会说训练、饮食等都很重要，但心理因素同样重要，甚至可能更为关键。你需要通过心理暗示让自己相信这些目标是可以实现的，这样你才会有动力去完成。

Is there something our human mind will never be able to comprehend well I sort of I guess the method is I mean yeah okay it's a bit of a reduction I it's between a lot there must be some it's between large number that you got I'm just that I was the first thing I came to mind so that but even broadly is there are we limb is there something about our mind that's we're going to be limited even with the help of mathematics well okay.

我们的思维是否存在一些永远无法真正理解的事物？我觉得是有的，虽然说法可能比较简化。我认为在很多情况下确实存在一些限制，特别是当涉及到非常庞大的数目时。这个念头在我脑海中第一个浮现，但从广义上说，即便借助数学，我们的思维仍然有可能存在某种限制。对吧。

I mean it's like how much augmentation are you willing like for example if I didn't even have pen and paper um like if I had no technology whatsoever okay so I'm not allowed blackboard pen and paper right you're already much more limited than you would be incredibly limited even language the English language is a technology it's uh it's one that's been very internalized so you're right they're really the the formulation of the problems in correct because there really is no longer a just a solo human.

这段话的意思是说，你愿意接受多少技术扩展？举个例子，如果我连笔和纸都没有，或者完全没有任何技术手段，比如连黑板、笔和纸都不用，你的能力就会受到了极大的限制。甚至语言本身，像英语这样的语言，也是种技术，已经被我们深深内化了。所以你说得对，问题的提出方式可能不太准确，因为实际上已经不可能存在完全孤立的人类个体了。

We're already augmented in extremely complicated intricate ways right yeah yeah we're already like a collective intelligence yes yes I guess so humanity plural has much more intelligence principally on his good days then then the individual humans put together it can all have less okay but um yeah so yeah math math math math the mathematical community plural is incredibly super intelligent uh entity um that uh no single human mathematician can can come close to to replicating.

我们已经在极为复杂的方式下得到了增强，对吧？是的，是的，我们已经像一个集体智能一样，是的，是的，我想是这样的。从整体上看，人类在状态好的时候具有的智慧远超单个个体的总和，但在状态不好的时候可能会少一些。数学，数学，数学，数学，数学界作为一个整体是一个无比聪明的实体，没有任何一个数学家能够接近这种智慧。

You see it a little bit on these like question analysis sites um uh so this mathematical flow which is the mathematical version of stackable flow and like sometimes you get like this very quick responses to very difficult questions from the community um and it is it's a pleasure to watch actually oh well as an expert I'm a fan spectator of that uh of that site just seeing the brilliance of the different people they the deaf analogous than people have and though the willingness to engage in the in the rigor and the nuance of the particular questions pretty cool to watch.

在一些问答分析网站上，您能看到这一点，嗯，比如说这里的数学流，这相当于数学版的Stack Overflow。社区有时候会对一些非常困难的问题给出快速的回应，这实际上是令人愉悦的。作为一个专家，我是该网站的忠实观众，欣赏不同人的才华，他们各具天赋。人们愿意参与到这些问题的严格性和细微差别中，真是非常有趣。

It's fun it's almost like just fun to watch uh what gives you hope about this whole thing we have going on human civilization I think uh yeah the um the younger generation is always like really creative and enthusiastic and and inventive um i it's a pleasure working with with uh with uh with young students um you know the uh the progress of science tells us that the problems that used to be really difficult can become extremely you know can become like trivial to solve you know i mean i you know it was like navigation you know I just was knowing where you were on the planet was this horrendous problem people people died um you know i lost fortunes. because so they couldn't navigate you know and we have devices in our pockets that do it automatically for us like is a completely solved problem you know so things that i've seen unfeedable for us now could be maybe just with homeic exercises with you yeah one of the things i find really sad about the finiteness of life is that i won't get to see all the cool things we create as a civilization you know that because it in the next hundred years two hundred years just imagine showing showing up in two hundred years.

这真是一种乐趣，观看这一切几乎就像是一种乐趣。那么，对于我们正在进行的这个人类文明，是什么让你充满希望呢？我认为，年轻一代总是非常有创造力、热情和发明力，与年轻学生一起工作真的是一种快乐。科学的进步告诉我们，以前非常困难的问题可以变得极易解决。比如说，曾经要知道自己在地球上的位置是一个巨大的难题，人们为此丧命，失去财富，因为他们无法进行导航。但现在，我们口袋里的设备可以自动为我们做这件事，导航已经完全不是问题了。因此，现在看似无法解决的问题，也许将来只需做一些家庭作业就能解决。我觉得生命有限这一点让人感到遗憾，因为我不能看到我们作为一个文明创造的所有酷炫东西。想象一下，再过一两百年，我们的世界会有多么不同。

yeah well already plenty has happened you know like if you could go back in time and and talk to you if you know teenage self or something you know i mean yeah yeah i'm just the internet and and now AI i mean it's like i get the they've been into they've been getting to internalize and say yeah of course uh and i can understand a voice and and give reason why you know slightly you can correct answers to any question but you know this was mind blowing even two years ago and in the moment it's hilarious to watch on the internet and so on the the drama uh people take everything for granted very quickly and then they we humans seem to entertain ourselves with drama well out of anything that's created somebody needs to take one opinion another person needs to take an opposite opinion argue with each other about it but when you look at the arc of things i mean it's just even in progress of robotics yeah just to take a step back and be like wow this is beautiful the way humans are able to create this yeah when they infrastructure and the culture is is healthy yeah the community of humans can be so much more intelligent and mature and rational the individuals within it.

嗯，确实，已经发生了很多事情。如果你能回到过去，与你十几岁的自己对话，你知道我的意思，是这样的。网络和现在的人工智能，真的让人觉得不可思议，你知道吗？这些已经被我们内化，成为理所当然的东西，比如理解语音和给出理由，能够对任何问题给出相对正确的答案。但你知道，就在两年前，这还是让人惊叹的，而在现在，网上的那些戏剧性的东西看起来让人觉得有趣。人们很快就把一切视为理所当然，然后我们人类似乎总是找点戏剧性来娱乐自己。无论创造出什么，总会有人持一种意见，而另一个人则持相反意见，然后彼此争论。但当你看待事情的发展轨迹时，即使只是机器人技术的进步，稍微退一步，就会感叹人类能够创造这些是多么美妙。当基础设施和文化状况良好时，人类的集体智慧、成熟和理性都能远超个体。

well one place i can always count on rationality is the comments section of your blog which i'm a fan of there's a lot of really smart people there and thank you of course for for putting those ideas out on the blog and it's i can't tell you how uh on or drem that you would spend your time with me today i was looking forward to this for a long time tear i'm a huge fan um you inspire me you inspire millions of people thank you so much for telling thank you it was a pleasure thanks for listening to this conversation with Terence Tao to support this podcast please check out our sponsors in the description or at lexfreedman.com slash sponsors and now let me leave you with some words from golele golele mathematics is a language with which god has written the universe thank you for listening and hope to see you next time.

翻译如下：我总是能够在你的博客评论区找到理性的声音，我是你的粉丝，那里的评论里有很多非常聪明的人。当然，也要感谢你在博客上分享那些想法。我无法用言语表达你今天抽时间和我交流有多么荣幸，我期待这一天已经很久了。我是你的超级粉丝，你激励着我，也激励了数百万人。非常感谢你告诉我的一切，这次交流非常愉快。感谢你收听这次和陶哲轩的对话。为了支持这个播客，请查看描述中的赞助商信息或访问 lexfreedman.com/sponsors。最后，让我引用一段关于数学的话来结束这次对话：“数学是上帝用来书写宇宙的语言。”感谢收听，希望下次再见。

Terence Tao: Hardest Problems in Mathematics, Physics &amp; the Future of AI | Lex Fridman Podcast #472

摘要

中英文字稿

Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472