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以下是与陶哲轩的对话，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，并在数学和物理学的众多领域做出了开创性贡献。这对我来说是莫大的荣幸，原因有很多，包括特里在我们所有互动中对我表现出的谦逊和善意。这对我意义重大。现在快速介绍一下各位赞助商。
The following is a conversation with Terence Tau, 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. And now a quick few second mention of each sponsor.
请在描述中或访问 lexfreedman.com/sponsors 查看他们。这是支持本播客的最佳方式。我们有团队协作工具 Notion、在线销售平台 Shopify、企业管理系统 NetSuite、电解质饮料 Element 以及健康品牌 AG1。她是我朋友中的智者。现在进入完整的广告播报环节，它们都集中在这里。
Check them out in the description or at lexfreedman.com/sponsors. It's the best way to support this podcast. We got Notion for teamwork, Shopify for selling stuff online, NetSuite for your business, Element for electrolytes, and the AG one for your health. She's wise in my friends. And now onto the full ad reads, they're all here in one place.
我确实会通过谈论一些我正在阅读或思考的随机话题来让广告变得有趣。但如果您跳过广告，也请务必查看赞助商。我喜欢他们的产品，或许您也会喜欢。如需联系我，请访问 laxtreatment.com/contact。好了。
I do try to make them interesting by talking about some random things I'm reading or thinking about. But if you skip, please still check out the sponsors. I enjoy their stuff, maybe you will too. To get in touch with me for whatever reason, go to laxtreatment.com/contact. Alright.
我们开始吧。本集由笔记和团队协作工具 Notion 赞助。我用 Notion 处理一切事务：个人笔记、制作播客、与他人协作，并通过 AI 超级增强所有这些功能，因为 Notion 在整合 AI 方面做得非常出色。有趣的是，在广泛采用书写和记录技术之前，尤其是在计算机出现之前，人类记忆的机制是怎样的。例如，您可以看看中世纪的僧侣，他们会使用现已广为人知的记忆技巧，如记忆宫殿、空间记忆技巧来背诵整本书籍。
Let's go. This episode is brought to you by Notion, a note taking and team collaboration tool. I use Notion for everything, for personal notes, for playing this podcast, for collaborating with other folks, and for super boosting all of those things with AI because Notion does a great job of integrating AI into the whole thing. You know what's fascinating is the mechanisms of human memory before we had widely adopted technologies and tools for writing and recording stuff, certainly before the computer. So you can look at medieval monks for example that would use the now well studied memory techniques, like the memory palace, the spatial memory techniques to memorize entire books.
这无疑是技术带来的影响：从谷歌搜索开始，发展到包括 Notion 在内的所有其他工具，我们将越来越多的记忆任务卸载给计算机。我认为这可能是件好事，因为它释放了我们更多的大脑来进行深度推理，即专注的专业深度挖掘或新闻记者式的思考，而不是记忆事实。不过，我确实认为，当你记忆大量内容时，会形成一种类似布拉特-柯尔模型的结构，从中会激发灵感并产生发现。所以我不确定。将大部分记忆任务卸载给机器可能会带来巨大代价，但这就是世界的发展方向。请访问 notion.comlex（全部小写）免费试用 Notion AI，立即体验 Notion AI 的强大功能。
That is certainly the effect of technology started by Google search and moving to all the other things like notion that we're offloading more and more and more of the task of memorization to the computers, which I think is probably a positive thing because it frees more of our brain to do deep reasoning, that's deep dive focused specialization or the journalist type of thinking versus memorizing facts. Although, I do think that there's a kind of brat curl model that's formed when you memorize a lot of things and from there, from inspiration arises discovery. So I don't know. It could be a great cost to offloading most of our memorization to the machines, but it is the way of the world. Try Notion AI for free when you go to notion.comlex, that's all lowercase notion.comlex to try the power of Notion AI today.
本集还由 Shopify 赞助，这是一个专为任何人在任何地方销售而设计的平台，提供美观的在线商店。我们的未来朋友中有很多机器人。展望遥远的未来，亚马逊仓库将拥有数百万机器人来搬运包裹。特斯拉机器人将遍布工厂、家庭、街道甚至咖啡店。所有这些就是我们的未来。
This episode is also brought to you by Shopify, a platform designed for anyone to sell anywhere with a great looking online store. Our future friends has a lot of robots in it. Looking into that distant future, have Amazon warehouses with millions of robots that move packages around. You have Tesla bots everywhere in the factories and in the home and on the streets and the baristas. All of that, that's our future.
目前，我们有像 Shopify 这样的平台在数字空间连接许多人。但越来越多地，在物理空间中将出现自动化、数字化、由 AI 驱动的人际连接。像许多未来一样，会有消极的一面，也会有积极的一面。而且像许多可能的未来一样，我们几乎无法阻止它的到来。我们所能做的就是引导它走向促进人类繁荣的方向。
Right now, have something like Shopify that connects a lot of humans in the digital space. But more and more, there will be a automated, digitized AI fueled connection between humans in the physical space. Like a lot of futures, there's going to be negative things and there's going to be positive things. And like a lot of possible futures, there's little we could do about stopping it. All we can do is steer it in the direction that enables human flourishing.
与其躲在恐惧中或散布恐慌，不如加入那些正在构建人类文明最佳发展轨迹的人群。总之，在shopify.com/lex注册每月1美元的试用期，注意全部小写。立即访问shopify.com/lex，将您的业务提升到新水平。本期节目也由NetSuite赞助，这是一款一体化云端业务管理系统。经营企业涉及许多杂乱环节，我不禁要问，也不禁思考，何时会出现像公司首席财务官一样的人工智能AGI。
Instead of hiding in fear or fear mongering, be part of the group of people that are building the best possible trajectory of human civilization. Anyway, sign up for a $1 per month trial period at shopify.com/lex, that's all lowercase. Go to shopify.com/lex to take your business to the next level today. This episode is also brought to you by NetSuite, an all in one cloud business management system. There's a lot of messy components to running a business and I must ask and I must wonder at which point there's going to be an AI AGI like CFO of a company.
一个处理大部分（若非全部）财务职责的人工智能代理，或承担NetSuite所有功能，到那时NetSuite是否会越来越多地利用AI完成这些任务。我认为它可能会将AI集成到工具中，但确实存在许多边缘情况需要人类智慧、基于多年经验的人类直觉来做出棘手决策。我猜测经营公司比人们意识到的要困难得多，但有很多文书类工作可以实现自动化、数字化、汇总整合，并作为人类决策的基础。总之，这就是我们的未来。在netsuite.com/lex下载首席财务官的人工智能与机器学习指南。
An AI agent that handles most if not all of the financial responsibilities or all of the things that NetSuite is doing at which point will NetSuite increasingly leverage AI for those tasks. I think probably it will integrate AI into its tooling, but I think there's a lot of edge cases that we need the the human wisdom, human intuition grounded in years of experience in order to make the tricky decision around the edge cases. I suspect that running a company is a lot more difficult than people realize, But there's a lot of sort of paperwork type stuff that could be automated, could be digitized, could be summarized, integrated, and used as a foundation for the said humans to make decisions. Anyway, that's our future. Download the CFO's guide to AI and machine learning at netsuite.com/lex.
网址是netsuite.com/lex。本期节目也由Element赞助，这是我日常饮用的零糖美味电解质冲剂。我经常沿河跑步，并遇到一些非常有趣的人。其中一位正在准备他的首场超级马拉松。我记得他说是100英里，这当然激发了我自己也一定要完成一场的想法。
That's netsuite.com/lex. This episode is also brought to you by Element, my daily zero sugar and delicious electrolyte mix. Now I run along the river often and get to meet some really interesting people. One of the the people I met was preparing for his first ultra marathon. I believe he said it was a 100 miles and that of course sparked to me the thought that I need for sure to to do one myself.
不久前，我计划与David Goggins一起做点什么，我认为我们俩之间完成某项疯狂体能壮举仍然在待办清单上。当然，对我而言疯狂的事对Goggins只是日常活动。但无论如何，我认为在体能领域、精神领域以及生活的所有领域挑战自己都很重要，而体育努力是挑战自我最清晰、结构最完善的方式之一，但挑战形式多种多样。写一本书，说实话，养育子女、婚姻、恋爱关系、友谊，所有这些如果你认真对待，全力以赴并做对，我认为那都是严峻的挑战。因为我们大多数人都没有做好准备。
Some time ago now, I was planning to do something with David Goggins and I think that's still on the sort of to do list between the two of us to do some crazy physical feat. Of course, the thing that is crazy for me is a daily activity for Goggins. But nevertheless, I think it's important in the physical domain, the mental domain, and all domains of life to challenge yourself and athletic endeavors is one of the most sort of crisp clear well structured way of challenging yourself, but there's all kinds of things. Writing a book, to be honest having kids and marriage and relationships and friendships, all of those if you take it seriously, if you go all in and do it right, I think that's a serious challenge. Because most of us are not prepared for it.
你一路学习，如果你拥有不断改进、个人成长并真正出色完成任务的严格反馈循环，我认为那无异于一场超级马拉松。总之，任意购买即可免费获赠样品包。请到drinkelement.com/lex尝试。最后，本期节目也由AG1赞助，这是一款支持改善健康和巅峰表现的一体化每日饮品。我每天都喝。
You learn along the way and if you have the rigorous feedback loop of improving constantly and growing as a person and really doing a great job of the thing, I think that might as well be an ultra marathon. Anyway, get a sample pack for free with any purchase. Try it to drinkelement.com/lex. And finally, this episode is also brought to you by AG one, an all in one daily drink to support better health and peak performance. I drink it every day.
我正在准备一场关于第三帝国毒品的对话。有趣的是，这是一种分析希特勒传记的方式。就是审视他全程摄入的东西，Norman Oller出色地分析了所有这一切，并以历史学家未曾触及的方式讲述了希特勒和第三帝国的故事。从非主流视角审视历史关键时刻总是令人欣喜。总之，我提到这一点是因为我认为希特勒有很多胃部问题，这就是他找医生的动机。
I'm preparing for a conversation on drugs in the third Reich. And funny enough, it's a kind of way to analyze Hitler's biography. Is to look at what he consumed throughout and Norman Oller does a great job of analyzing all of that and tells the story of Hitler and the third Reich in a way that hasn't really been touched by historians before. It's always nice to look at key moments in history through a perspective that's not often taken. Anyway, I mentioned that because I think Hitler had a lot of stomach problems and so that was the motivation for getting a doctor.
那位医生最终给他用了各种药物，但医生通过提供益生菌赢得了希特勒的信任，这在当时算是革命性的事物。这确实帮助解决了希特勒所患的胃部问题。所有这些都提醒我们，战争是由人类发动的，而人类是生物系统，生物系统需要燃料、补充剂等各类东西，摄入体内的物质会影响短期和长期表现，对希特勒而言，直到他在柏林地堡的最后日子，他所服用的各种药物鸡尾酒都是如此。所以我觉得我把自己带到了某个深处。
The doctor that eventually would fill him up with all kinds of drugs, but the doctor earned Hitler's trust by giving him probiotics, which is a kind of revolutionary thing at the time. And so that really helped deal with whatever stomach issues that Hitler was having. All of that is a reminder that war is waged by humans and humans are biological systems and biological systems require fuel and supplements and all of that kind of stuff, and depending on what you put in your body will affect your performance in the short term and the long term with meth. That's true with Hitler to his last days in the bunker in Berlin, All the cocktail of drugs that he was taking. So I think I got myself somewhere deep.
我不确定该如何摆脱这个困境。这值得进行数小时的对话，而不是仅仅几秒钟的提及。但没错，这一切都是源于我对AG1的思考以及我对它的热爱。感谢你收听这个节目，并跟随这些广告朗读一起经历这段疯狂的旅程。总之，当你注册drinkag1.com/lex时，AG1会赠送你一个月的鱼油供应。
I'm not sure how to get out of this. It deserves a multi hour conversation versus a few seconds of mention. But yeah, all of that was sparked by my thinking of AG one and how much I love it. I appreciate that you're listening to this and coming along for the wild journey that these ad reads are. Anyway, a g one will give you a one month supply of fish oil when you sign up at drinkag1.com/lex.
这里是Lex Friedman播客。如需支持，请查看描述中的赞助商或访问lexfreeman.com/sponsors。现在，亲爱的朋友们，有请陶哲轩。你遇到的第一个真正困难的研究级数学问题是什么？也许是让你感到棘手的问题。
This is the Lex Friedman podcast. To support it, please check out our sponsors in the description or at lexfreeman.com/sponsors. And now, dear friends, here's Terence Tao. What was the first really difficult research level math problem that you encountered? One that gave you pause maybe.
嗯，我的意思是，
Well, I mean,
在本科教育中，你会了解到那些真正困难、几乎不可能解决的问题，比如黎曼猜想或素数猜想。你可以让问题变得任意困难，但那并不真正算是一个问题。事实上，甚至有些问题我们知道是无法解决的。真正有趣的是那些处于我们能轻松解决和完全无望之间的边界问题。也就是那些现有技术能完成90%的工作，而你只需要攻克剩下10%的问题。
in your undergraduate education, you you learn about the really hard impossible problems, like the Riemann hypothesis between primes conjecture. You can make problems arbitrarily difficult, but that's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just to the on the boundary between what we can do fairly easily and what are hopeless. But what are problems where, like, existing techniques can do, like, 90% of the job, and then you you just need that remaining 10%.
我认为作为博士生，Kakeya问题确实引起了我的注意，而且它最近刚被解决。这个问题在我早期研究中投入了很多精力。历史上，它源自日本数学家Kakeya Soji在1918年左右提出的一个小谜题。这个谜题是：你在平面上有一根针。可以想象成在道路上开车之类的场景。
I think as a PhD student, the Kakaya 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, Soji Kakaya, in, like, 1918 or so. So the puzzle is that you you you have a needle in on the plane. Think of like like a like driving on a like on on a road or something.
你想要执行一个U型转弯。你想让针调头，但希望使用尽可能小的空间。所以你希望用最少的面积来完成调头。但这根针具有无限的可操作性。你可以想象围绕它的中心旋转——这是一根单位长度的针。
And you you wanted to execute a U-turn. You wanna turn the needle around, but you wanna do it in as little space as possible. So you wanna use as little area in order to turn it around. So but the needle is infinitely maneuverable. So you can imagine just spinning it around its it's a unit needle.
你可以围绕它的中心旋转，我认为这样需要的是一个面积为π/4的圆盘。或者你可以做一个三点调头，就像驾校教的那样，这实际上只需要π/8的面积。所以它比旋转更高效一些。有一段时间，人们认为这是最有效的调头方式。但Vasikovich证明，实际上你可以用任意小的面积让针调头。
You can spin it around its center, and I think that gives you a disc of of area, I think, pi over four. Or you can do a three point u-turn, which is what we teach people in in the driving schools to do, and that actually takes area pi over eight. So it's 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 Vasikovich showed that, in fact, you could actually turn the needle around using as little area as you wanted.
所以第0.001点，有一种非常精巧的多重来回U型转弯操作，你可以让指针完全调转方向。在这个过程中，它会经过每一个中间方向。
So point zero zero one, there was some really fancy multi back and forth u u-turn thing that you could you could do that that you could turn the needle around. And in so doing, it would pass through every intermediate direction.
这是在二维平面中吗？
Is this in the two dimensional plane?
这是在二维平面中。是的。所以我们在二维中理解所有情况。那么接下来的问题是三维中会发生什么。假设，比如哈勃太空望远镜在太空中，你想要观测宇宙中的每一颗恒星。
This is in the two dimensional plane. Yeah. 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.
所以你需要旋转望远镜以指向每一个方向。这里有个不现实的部分：假设空间非常宝贵（实际上完全不是），你希望占用尽可能小的体积来旋转你的指针，以便看到天空中的每一颗恒星。你需要多小的体积才能做到这一点？
So you want to rotate the telescope to reach every single direction. And here's the 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 your needle around in order to see every single star in the sky. How small a volume do you need to do that?
你可以修改贝西科维奇的构造。如果你的望远镜厚度为零，那么你可以使用任意小的体积。这是二维构造的一个简单修改。但问题是，如果你的望远镜不是零厚度，而是非常非常薄，有一定厚度δ，那么作为δ的函数，能够看到所有方向所需的最小体积是多少？随着δ变小，随着指针变薄，体积应该减小。
And so you can modify Bezikovitch's construction. And so if your telescope has zero thickness, then you you can use as little volume as you need. That's a simple modification of the two dimensional construction. But the question is that if your telescope is not zero thickness, but 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 the needle gets gets thinner, the volume should go down.
但它减少的速度有多快？猜想是它减少得非常非常缓慢，大致上是对数级的，经过大量工作后这一点得到了证明。这看起来像是一个谜题。为什么它有趣？结果发现它出人意料地与偏微分方程、数论、几何学、组合数学中的许多问题相关联。
But but how fast does it go down? And the conjecture was that it goes down very, very slowly, like logarithmic, roughly speaking, and that was proved after a lot of work. So this seems like a puzzle. Why is it interesting? So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, combinatorics.
例如，在波传播中，你搅动一些水，产生水波，它们朝各个方向传播。但波同时表现出粒子和波的行为特性。所以你可以有所谓的波包，这是一种在空间上非常局域化、随时间朝特定方向移动的波。如果你在空间和时间中绘制它，它占据的区域看起来像一个管状结构。
For example, in in wave propagation, you splash some some water around. You create water waves, and they they travel in various directions. But waves exhibit both both particle and wave type behavior. So you can have what's called a wave packet, which is like a a very localized wave that is localized in space and moving a certain direction in time. And so if you plot it into space and time, it occupies a region which looks like a tube.
所以可能发生的情况是，一个最初非常分散的波，最终会在某个时间点全部汇聚到一个点上。就像你可以想象把一颗鹅卵石扔进池塘，涟漪会扩散开来。但如果你将这个过程时间反演——而波动方程是时间可逆的——你就可以想象涟漪汇聚到一个点上，然后产生一个大水花，甚至可能形成一个奇点。这是可能实现的。从几何角度看，这总是涉及到光线的存在。
And so what can happen is that you can have a wave which initially is very dispersed, but it all comes it all focuses at a single point later in time. Like, you can imagine dropping a pebble into a pond and ripple spread out. But then if you time reverse that that that scenario and the equations are wave motion are time reversible, you can imagine ripples that are converging to a single point, and then a a big splash occurs, maybe even a singularity. And so it's possible to do that. And geometric, what's going on is that there's always a light rays.
比如，如果这个波代表光，你可以把这个波想象成所有以光速运动的光子的叠加。它们都沿着这些光线传播，并且都汇聚到这一个点上。所以你可以让一个非常分散的波在时空中的某一点聚焦成一个非常集中的波，然后它又会再次散开分离。但假如这个猜想有一个否定解，那就意味着存在一种非常高效的方式，可以将指向不同方向的管道压缩到一个非常非常小的区域内。那么你也就能创造出一些最初非常分散的波，它们不仅会汇聚到单个点，还会在时空中有大量的集中点。
So, like, if if 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 very dispersed wave focus into a very concentrated wave at one point in space and time, but then it defocuses again and it it separates. But potentially, if the conjecture had a negative solution so what that meant is that there's there's a very efficient way to pack tubes pointing in different directions to a very, very narrow region of of of very narrow volume. Then you would also be able to create waves that start out some 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.
（爆发）现象，这些波的振幅变得如此之大，以至于它们所遵循的物理定律不再是波动方程，而是更复杂的非线性方程。因此在数学物理中，我们非常关注某些波动方程是否稳定，是否会产生这些奇点。有一个著名的问题叫做纳维-斯托克斯正则性问题。纳维-斯托克斯方程是描述不可压缩流体（如水）流动的方程。这个问题问的是，如果你从一个平滑的水流速场开始，它是否可能在某些点集中到速度变成无限大的程度？
And you could create what's called a blow up where these waves their amplitude becomes 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 in in in wave equations are stable or not, whether they can create these singularities. There's a famous problem called the Navier Stokes regularity problem. So the Navier Stokes equations, equations that govern the fluid flow for incompressible fluids like water. The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that, like, the velocity become infinite at some point?
这就叫做奇点。我们在现实生活中看不到这种情况。你知道的，如果你在浴缸里玩水，水不会爆炸或者以光速飞溅什么的。但理论上这是可能的。事实上，近年来共识已经逐渐转向认为，对于某些非常特殊的初始配置（比如水的特定初始状态），确实可能形成奇点。
That's called a singularity. We don't see that in real life. You know? If you splash around water on the on the bathtub, it won't explode on you or or have have water leaving a speed of light or anything, but potentially, it is possible. And in fact, in recent years, the the consensus has has drifted towards the the the belief that that in fact, for certain very special initial con configurations of of say water, that singularities can form.
人们目前还未能真正证实这一点。克莱数学研究所有七个千禧年大奖难题，解决其中任何一个问题都能获得一百万美元的奖金，这就是其中之一。这七个问题中，只有一个已经被解决，就是佩雷尔曼解决的庞加莱猜想。卡克亚猜想与纳维-斯托克斯问题没有直接关系，但理解它会帮助我们理解波集中等某些方面，这可能会间接帮助我们更好地理解纳维-斯托克斯问题。
People have not yet been able to 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. They the Poincare Conjecture by Perlman. So the Kakarya conjecture is not directly 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.
你能谈谈纳维-斯托克斯问题吗？就是你说的存在性和光滑性，千禧年大奖难题。你在这个问题上取得了很大进展。2016年，你发表了一篇关于三维平均纳维-斯托克斯方程有限时间爆发的论文。
Can you speak to the Navier Stokes? So the existence and smoothness, like you said, millennial price problem. Right. You've 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.
是的。
Right.
所以我们在试图弄清楚这个东西通常不会爆炸。对吧。但我们能确定它永远不会爆炸吗？
So so we're trying to figure out if this thing usually doesn't blow up. Right. But can we say for sure it never blows up?
对。是的。所以，这确实就是那个价值百万美元的问题。是的。这就是数学家与几乎所有其他人的区别所在。
Right. Yeah. So, yeah, that is literally the the the million dollar question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else.
就像，如果如果如果某事在99.99%的情况下都成立，对大多数人来说，你知道，对大多数事情来说，这就足够了。但数学家是少数真正关心是否每一个，比如100%，真正100%的所有情况都被覆盖的人。所以大多数流体大多数时候，水不会爆炸。但你能设计出一个非常特殊的初始状态来做到这一点吗？
Like, if if if something holds 99.99% of the time, that's good enough for most, you know, for for for most things. But mathematicians are one of few people who really care about whether every like, 100% really 100% of all situations are covered by by yeah. So most fluids most of the time, water does not blow up. But could you design a very special initial state that does this?
也许我们应该说明这是一组在流体动力学领域起支配作用的方程。是的。试图理解流体的行为，结果发现它实际上是一个非常复，你知道，流体是...是的。极其复杂的东西，很难建模。
And maybe we should say that this is a this is a set of equations that govern in the field of fluid dynamics. Yes. Trying to understand how fluid behaves, it's actually turns out to be a really comp you know, fluid is Yeah. Extremely complicated thing to try to model.
是的。所以它具有实际重要性。这个克莱奖问题涉及所谓的不可压缩纳维-斯托克斯方程，它支配着像水这样的东西。还有所谓的可压缩纳维-斯托克斯方程，它支配着像空气这样的东西。这对天气预报尤其重要。
Yeah. So it has practical importance. So this clay price 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. And that's particularly important for weather prediction.
天气预报，它做了大量的计算流体动力学。其中很多实际上只是尽力求解纳维-斯托克斯方程。同时，收集大量数据以便他们能够初始化方程。有很多活动部件。所以这实际上是一个非常重要的问题。
Weather prediction, it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve Navier Stokes equations as as best they can. Also, gathering a lot of data so that they can get they can initialize the equation. There's a lot of moving parts. So it's very important problem practically.
为什么证明关于这组方程的一般性结论，比如它不会爆炸，如此困难？
Why is it difficult to prove general things about these set of equations like it not not blowing up?
简而言之就是麦克斯韦妖。麦克斯韦妖是热力学中的一个概念。比如，如果你有一个装有氧气和氮气两种气体的盒子，可能一开始氧气在一侧，氮气在另一侧，但它们之间没有隔板。对吧？然后它们就会混合。
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 a oxygen and nitrogen, and maybe you start with all the oxygen one side and nitrogen the other side, but there's no barrier between them. Right? Then they will mix.
它们应该保持混合状态。对吧？没有理由它们会重新分离。但原则上，由于它们之间的所有碰撞，可能存在某种奇怪的共谋，就像有一个叫做麦克斯韦妖的微观恶魔，每次氧气和氮气原子碰撞时，它们会以某种方式反弹，使得氧气逐渐漂移到一侧，而氮气则移动到另一侧。你可能会得到一个极其不可能的配置出现，这种情况我们从未见过，从统计学上来说，这是极其不可能的。
And they should stay mixed. Right? There's there's no reason why they should unmix. But in principle, because of all the collisions between them, there could be some sort of weird conspiracy that that like, maybe there's a microscopic demon called Maxwell's demon that will every time oxygen and nitrogen atom collide, they will bounce off in such a way that the oxygen sort of drifts onto one side and then nitrogen goes to the other. And you could have an extremely improbable configuration emerge, which we never see, and and which we statistically, it's extremely unlikely.
但从数学上来说，这种情况是可能发生的，我们不能排除这种可能性。这种情况在数学中经常出现。一个基本例子是圆周率π的数字，3.14159等等。这些数字看起来没有规律，我们相信它们确实没有规律。从长期来看，你应该看到1、2、3出现的次数与4、5、6一样多。
But mathematically, it's possible that this can happen, and we can't rule that out. And this is a situation that shows up a lot in mathematics. Basic example is the digits of pi, 3.14159 and so forth. The digits look like they have no pattern, and we believe they have no pattern. On the long term, you should see as many ones and twos and threes as fours and fives and sixes.
π的数字不应该偏爱某个数字，比如说7比8更常出现。但也许π的数字中有某种恶魔，每当你计算更多位数时，它就会偏向某个数字。这是一种不应该发生的共谋，没有理由会发生，但以我们目前的技术无法证明这一点。好吧。
There should be no preference in the digits of pie to favor, let's say, seven over eight. But maybe there is some demon in the digits of pie that that, like, every time you compute more more digits, it biases one digit to another. And this is a conspiracy that should not happen. There's no reason it it should happen, but there's there's there's no way to prove it with our current technology. Okay.
回到纳维-斯托克斯方程，流体具有一定量的能量。由于流体在运动，能量会在各处传输，而且水也具有粘性。所以如果能量分散在许多不同位置，流体的自然粘性就会耗散能量，最终能量会归零。这就是我们在实际水实验中观察到的情况。我明白了。
So getting back to Navier Stokes, a fluid has a certain amount of energy. And because the fluid is in motion, the energy gets transported around, and water is also viscous. 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 it will it will go to zero. And this is what happens in when we actually experiment with water. I get it.
你搅动水时，会产生一些湍流和波浪等等，但最终它会平静下来，振幅越小，速度越低，就会变得越平静。但可能存在某种恶魔，不断将流体的能量推向越来越小的尺度。它会移动得越来越快，在更快的速度下，有效粘性相对较小。因此可能会发生所谓的自相似团块情景，流体的能量从某个大尺度开始，然后全部转移到流体的一个较小区域，接着以更快的速度转移到更小的区域，依此类推。每次这样做所需的时间可能是前一次的一半。
You splash around, there's there's some turbulence and and waves and so forth, but eventually, it it settles down, and and and the the lower the amplitude, the the smaller velocity, the the more calm it gets. But potentially, there is some sort of demon that keeps pushing the the energy of the fluid into a smaller and smaller scale. And it will move faster and faster, and at faster speeds, the effective viscosity is relatively less. And so it could happen that that it it creates some sort of what's called a self similar blob scenario where, you know, the end of your fluid starts off at some large scale, and then it all sort of transfers energy into a smaller region of of of the fluid, which then at a much faster rate moves into an even smaller region and so forth. And and each time it does this, it takes maybe half as as long as as the previous one.
然后，是的，你实际上可以在有限时间内将所有能量汇聚到一点。这就是所谓的Fondheim爆炸。在实践中，这不会发生。所以水被称为湍流。确实，如果你有一个大水涡，它倾向于分解成更小的水涡，但不会将所有能量从一个大水涡转移到一个小的水涡中。
And then, yeah, you you could you could actually converge to to all the energy concentrating at one point in a a finite amount of time. And that that that's that's an hour he's got Fondheim 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, but it won't transfer all this energy from one big eddy into one small eddy.
它会分裂成大概三四个涡旋。然后这些涡旋又会各自分裂成大概三四个更小的涡旋。这样能量就会分散到粘性能够控制的程度。但如果它能以某种方式将所有能量集中起来，保持在一起，并且速度足够快，让粘性效应来不及平息一切，那么这种能量团就可能出现。所以有论文声称，哦，你只需要考虑能量守恒并仔细运用粘性，就能控制住一切——不仅适用于纳维-斯托克斯方程，还适用于许多类似的方程。
It will transfer into maybe three or four. And then those must split up into maybe three or four small eddies of their own. And so the energy gets dispersed to the point where the viscosity can can then keep the 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 calm everything down, then this blob can occur. So there are papers who had claimed that, oh, you just need to take into account conservation energy and just carefully use the viscosity, and you can keep everything under control for not just in Navier Stokes, but for many, many types of equations like this.
因此过去有很多尝试想要获得所谓的纳维-斯托克斯全局正则性，这与Vanatten Bloor相反，即速度保持光滑。但所有这些尝试都失败了。总是存在一些符号错误或细微的失误，无法挽救。所以我感兴趣的是试图解释为什么我们无法证伪有限时间爆破。我无法对实际的流体方程做到这一点，因为它们太复杂了。
And so in the past, there have been many attempts to try to obtain what's called global regularity for Navier Stokes, which is the opposite of Vanatten Bloor, that velocity stays smooth. And it all failed. There was always some sign error or some subtle mistake, and and it it couldn't be salvaged. So what I was interested in doing was trying to explain why we were not able to disprove, part of time blow up. I couldn't do it for the actual equations of fluids, which were too complicated.
但如果我能平均化纳维-斯托克斯的运动方程，基本上就是如果我能够关闭水相互作用的某些方式，只保留我想要的那些。具体来说，如果有一个流体能够将能量从大涡旋传递到这个或那个小涡旋，我会关闭传递能量到这个涡旋的通道，只将其导向这个更小的涡旋，同时仍保持低能量集中度。
But if I could average the equations of motion of Navisokes, so, basically, if if if I could turn off certain types of of ways in which water interacts and and only keep the ones that I want. So in particular, if if there's a fluid and it could transfer its energy from a large eddy into this small eddy or this other small eddy, I would turn off the energy channel that would transfer energy to this this one and and direct it only into this smaller eddy while still preserving the low concentration energy.
所以你在试图制造一个爆破？
So you're trying to make a blow up?
是的。是的。所以我基本上是通过改变物理定律来设计一个爆破，这是数学家被允许做的事情。我们可以改变方程。
Yeah. Yeah. So I I I basically engineer a blow up by changing laws of physics, is one thing that mathematicians are allowed to do. We can change the equation.
这如何帮助你更接近某个证明呢？
How does that help you get closer to the proof of something?
对。这提供了数学中所谓的障碍。我所做的是，基本上，如果我关闭方程的某些部分——通常当你关闭某些相互作用时，会使其非线性减弱，从而更规则且更不容易爆破。但我发现通过关闭一组精心设计的相互作用，我可以迫使所有能量在有限时间内爆破。这意味着如果你想证明实际纳维-斯托克斯方程的全局正则性，你必须利用真实方程的某些特性，而这些特性是我的人工方程所不具备的。
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, so 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 of of of of interactions, I could force all the energy to blow in finite time. So what that means is that if you wanted to prove global regularity for Navier Stokes for the actual equation, you had you must use some feature of the true equation, which which my artificial equation does not satisfy.
所以它排除了某些特定的方法。数学的关键在于，不仅仅是找到有效的方法并应用它，还需要避免使用那些无效的方法。对于真正困难的问题，通常有几十种可能适用的解决方式，但只有通过大量经验后，你才会意识到这些方法根本行不通。因此，针对相近问题的反例能够排除这些方法。这为你节省了大量时间，因为你不会在那些现在知道永远不可能奏效的事情上浪费精力。
So it it rules out certain certain approaches. So the thing about math is is is it's not just about finding know, taking a technique that is gonna work and applying it, but you you need to not take the techniques that don't work. 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 it saves you a lot of time because you you you're not wasting energy on on things that you now know cannot possibly ever work.
这与流体动力学的那个特定问题有多深的联系，还是只是你在数学中建立的一些更普遍的直觉？对。是的。我的技术所利用的关键现象被称为超临界性。在偏微分方程中，这些方程通常像是不同力量之间的拉锯战。
How deeply connected is it to that specific problem of fluid dynamics or just some more general intuition you build up about mathematics? Right. Yeah. So the key phenomenon that was my my technique exploits is what's called supercriticality. So in partial differential equations, often these equations are like a tug of war between different forces.
在纳维-斯托克斯方程中，有来自粘性的耗散力，这一点被很好地理解。它是线性的，能够使情况平静下来。如果只有粘性，那么就不会发生任何糟糕的事情。但还有输运作用，因为流体在运动，能量可以从空间的一个位置被输运到其他位置。
So in Navier Stokes, there's the dissipation force coming from viscosity, and it's very well understood. It's linear. It calms things down. If if viscosity was all there was, then then nothing bad would ever happen. But there's also transport that that energy from in one location of space can get transported because the fluid is in motion to to other locations.
这是一个非线性效应，它导致了所有的问题。所以在纳维-斯托克斯方程中有这两个相互竞争的项：耗散项和输运项。如果耗散项占主导，如果它很大，那么基本上你会得到正则性。而如果输运项占主导，那么我们就不知道会发生什么。这是一个非常非线性的情况。
And that's a nonlinear effect, and that causes all the all the problems. So there are these two competing terms in the Navier Stokes equation, the dissipation term and the transport term. If the dissipation term dominates, if it's if it's large, then basically you get regularity. And if, if the transport term dominates, then, then we don't know what's going on. It's a very nonlinear situation.
它是不可预测的，是湍流的。所以有时这些力量在小尺度上平衡，但在大尺度上不平衡，或者反之亦然。因此纳维-斯托克斯被称为是超临界的。在越来越小的尺度上，输运项远比粘性项强。
It's unpredictable. It's turbulent. So sometimes these forces are in balance at small scales, but not in balance at large scales or or vice versa. So Navier Stokes is what's called supercritical. So at at smaller and smaller scales, the transport terms are much stronger than the viscosity terms.
粘性项是使情况平静下来的因素，所以这就是问题困难的原因。在二维情况下，苏联数学家拉迪任斯卡娅在六十年代证明了二维情况下不会发生爆破。正如你提到的，纳维-斯托克斯方程被称为是临界的。即使在非常非常小的尺度上，输运效应和粘性效应的强度大致相同。我们有很多技术来处理临界和亚临界方程并证明正则性。
So the viscosity terms are things that calm things down, and so this is this is why the problem is hard. In two dimensions so the Soviet mathematician, Ladyshynskaya, she, in the sixties, showed in two dimensions, there was no blow up. And as you'd mentioned, the Navier Stokes equation is what's called critical. The effect of transport and the effect of viscosity are about the same strength even at very, very small scales. And we have a lot of technology to handle critical and also subcritical equations and prove regularity.
但对于超临界方程，情况就不明确了。我做了很多工作，随后也有大量后续研究表明，对于许多其他类型的超临界方程，你可以构造出各种爆破例子。一旦非线性效应在小尺度上主导了线性效应，各种糟糕的事情都可能发生。所以这条研究线的主要见解之一就是，超临界性与临界性和亚临界性之间的区别非常重要。我的意思是，这是一个关键的质量特征，它区分了某些方程是友好、可预测的，比如行星运动。
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. So this is sort of one of the main insights of this this line of work is that supercriticality versus criticality and subcriticality, this this makes a big difference. I mean, that that's a key qualitative feature that distinguishes some equations for being sort of nice and predictable and and, you know, like like planetary motion.
我的意思是，有些方程可以预测数百万年或者至少数千年的情况。再说一次，这其实不是问题，但我们无法预测未来两周以上的天气是有原因的，因为它是一个超临界方程。在非常精细的尺度上发生了许多非常奇怪的事情。
I mean, there's there's certain equations that that you can predict for millions of years and or or thousands at least. Again, it's not really a problem, but but there's a reason why we can't predict the weather past two weeks into the future because it's a supercritical equation. Lots of really strange things are going on at at very fine scales.
所以每当存在巨大的非线性来源时，是的。这会给预测将要发生的事情带来巨大问题。
So whenever there is some huge source of nonlinearity Yeah. That can create a huge problem for predicting what's gonna happen.
是的。而且如果非线性在某种程度上在更小的尺度上越来越显著和有趣。我的意思是，有很多方程是非线性的，但在许多方程中，你可以通过整体来近似处理。例如行星运动，如果你想了解月球或火星的轨道，你并不真正需要其他微观结构，比如月球的地震学或者质量是如何精确分布的。你基本上可以把这些行星近似为质点，重要的是它们的聚合行为。
Yeah. And if nonlinearity is somehow more and more featured and interesting at at small scales. I mean, there's there's many equations that are nonlinear, but in many in in many equations, you can approximate things by the bulk. So for example, planetary motion, you know, if you wanted to understand the orbit of the moon or Mars or something, you don't really need the microstructure of other like the seismology of the moon or or, like, exactly how the mass is distributed. You just basically you can also approximate these planets by point masses, and it's just the aggregate behavior is important.
但如果你想模拟像天气这样的流体，你不能简单地说在洛杉矶温度是这样，风速是这样。对于超临界方程，最精细的配置确实非常重要。
But if you wanna model a fluid like like the weather, you can't just say, in Los Angeles, the temperature is this, the wind speed is this. For supercritical equations, the finest confirmation is is really important.
如果我们能稍微多讨论一下纳维-斯托克斯方程。你之前提到，也许可以这样描述：解决它或负面解决它的方法之一是构建一种液体计算机。然后表明计算理论中的停机问题对流体动力学有影响。对吧。
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 computation theory has consequences for fluid dynamics. So Right.
用那种方式展示。你能描述一下这个想法吗？
Show it in that way. Can you describe this this idea?
是的。这个想法源于构建这个平均方程的工作，那个方程发散了。作为我不得不这样做的一部分，有一种天真的方法。你只是不断推进。每次你在一个尺度上获得能量，你就尽可能快地立即将其推到下一个尺度。
Yeah. So this came out of of this work of constructing the this this this average equation that that blew up. So one as as part of how I had to to do this so there's sort this naive way to do it. You you you just keep pushing. Every time you you get energy at one scale, you you push it immediately to the next scale as as fast as possible.
这是一种比较天真的强制爆破方式。结果发现在五维及更高维度中，这种方法确实有效。但在三维空间中，我发现了一个有趣的现象：如果你不断改变物理定律，试图将能量推向越来越小的尺度，能量反而会开始同时分散到多个尺度上。你在某个尺度上有能量，将其推向下一个尺度，但一旦进入新尺度，你又会继续推向更小尺度，而之前尺度的能量仍有残留。
This is sort of the naive way to to to to force blow up. It turns out in five and higher dimensions, this works. But in three dimensions, there was this funny phenomenon that I discovered that if you if you keep if if you change laws of physics, you just always keep trying to push the energy into smaller smaller scales. What happens is that the energy starts getting spread out into multi many scales at once. So you you have energy at one scale, you're pushing it into the next scale, and then soon as it enters that scale, you also push to the next scale, but there's still some energy left over from the previous scale.
你试图同时处理所有尺度，这会使能量过度分散。结果就是使得粘性效应有机可乘，最终将所有能量阻尼消耗。因此这种定向传递实际上行不通。其他作者的论文也证实了三维空间中的这一现象。所以我需要设计一种延迟机制，类似于气闸的工作原理。
You're trying to do everything at once, and this spreads out the energy too much. And then it turns out that that it makes it vulnerable for viscosity to come in and actually just damp out everything. So so it turns out this this directed motion doesn't doesn't actually work. There was a separate paper by some other authors that actually showed this in three dimensions. So what I needed was to program a delay, so kinda like airlocks.
我需要构建一个方程，让流体先在某个尺度上活动，将能量推向下一个尺度后，能量会暂存该尺度，直到所有来自更大尺度的能量完全转移。只有在全部能量注入后，才会开启下一级闸门继续传递。通过这种方式，能量就能逐级推进，始终保持单尺度局部化，从而有效抵抗粘性耗散效应。
So I needed an equation which would start with 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 from the larger scale got transferred. And only after you pushed all the energy in, then you sort of open the next gate, and and then you you push that in as well. So by doing that, it kind of the energy inches forward scale by scale in such a way that it's always localized at one scale at a time. And then it can resist the effects of viscosity because it's it's not dispersed.
为实现这个目标，我不得不构造一个相当复杂的非线性系统。其原理类似于电子电路设计——这里要感谢我妻子，她作为电子工程师受过专业训练。她常说起设计电路的经历：若需要实现特定功能（比如让指示灯闪烁），就需要用电容、电阻等基础元件搭建电路图，通过图纸就能直观判断电流的走向与中断时机。
So in order to make that happen, yeah, I had to construct a rather complicated nonlinearity. And it was basically like, you know, like, it was constructed like an electronic circuit. So I I actually thank my wife for this because she was trained as a electrical engineer. And, you know, she she talked about, you know, she had to design circuits and so forth. And, you know, if you if you want a circuit that does a certain thing, like, maybe have a light that that flashes on and then turns off and then on and then off, you can build it from from more primitive components, you know, capacitors and resistors and so forth, and you have to build a diagram.
我知道如何用数学模拟基础电子元件（如电阻、电容），并将它们组合成能按序开启闸门的系统——当时钟达到阈值时自动关闭。这就像鲁布·戈德堡机械的数学版，最终成功了。我意识到，若能在实际方程中实现类似计算（想象蒸汽朋克风格的水计算器），现代计算机依赖电子在微线路中传导交互，而我们可以用水脉冲以特定速度运动。
And you you and these diagrams, you can you can sort of follow with your eyeballs and say, oh, yeah. The the the current will pull up here, and then it will stop, and then it will do that. So I knew how to build the analog of basic electronic components, you know, like resistors and capacitors and so forth, and and I would I would stack them together in such in such a way that that I would create something that would open one gate, and then there'll be a clock that would and then once the clock hits the threshold, it would close it. They're kind of a Rube Goldberg type machine, but described mathematically, and this ended up working. 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 steampunk, it's really waterpunk type of thing where you know, so modern computers are electronic.
两种水团构型可对应比特位的0和1。当两股水流碰撞时，会产生新构型，实现与门或或门的逻辑功能——输出结果完全由输入决定。将这些逻辑单元串联后，或许能构建出图灵机，进而打造完全由水构成的计算机。有了计算机，甚至可能发展出水力机器人技术。
You know, they they they're they're powered by by electrons passing through very tiny wires and interacting with other other electrons and and so forth. But instead of electrons, you you can imagine these pulses of of water moving a certain velocity, and maybe it's they're two different configurations corresponding to a bit being up or down. Probably that if you had two of these moving bodies of water collide, they would come out with some new configuration, which is which would be something like an and gate or or gate, you know, that if if the the the output would depend in a very predictable way on on the inputs. And, like, you could chain these together and maybe create a Turing machine, and and then you could you have computers, which are made completely out of water. And if you have computers, then maybe you can do robotics.
最终可以制造出流体版的冯·诺依曼机器。冯·诺依曼曾提出：若要殖民火星，运输人员和设备的成本极高；但若送一台能采矿、冶炼、自我复制的机器到火星，就能逐步殖民整个星球。如果我们能造出这样的流体机器……
So, you know, hydraulics and so forth. And so you could create some machine, which is basically a fluid analog of what's called a Von Neumann machine. So Von Neumann proposed, if you wanna colonize Mars, the sheer cost of transporting people and 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 raw materials, smelt 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 yeah.
所以它是一个流体机器人。好吧。它的作用，它存在的目的，是被编程为在某种冷却状态下创建一个更小版本的自己。它暂时还不会启动。
So it's it's it's a it's a it's a fluid robot. Okay. And what it would do, it's it's purpose in life. It's programmed so that it would create a smaller version of itself in some sort of cold state. It it wouldn't start just yet.
一旦准备就绪，大型机器人配置的水会将所有能量转移到这个更小的配置中，然后关机。然后我会自行清理。剩下的就是这个最新状态，它会启动并做同样的事情，但更小更快。然后这个方程具有某种缩放对称性。
Once it's ready, the big robot configuration water would transfer all all its energy into this the smaller configuration and then power down. K. And then I I I clean it myself up. And then what's left is this newest state, which would then turn on and do the same thing, but smaller and faster. And then the equation has a certain scaling symmetry.
一旦做到这一点，它就可以不断迭代。所以这在原则上会为实际的纳维-斯托克斯方程创建一个模糊概念，而这就是我为这个平均纳维-斯托克斯方程所完成的。所以它提供了解决这个问题的路线图。但这只是空想，因为要实现这个目标还缺少太多东西。我无法创建这些基本逻辑门。
Once you do that, it can just keep iterating. So this in principle would create a blurb for the actual Navier Stokes, and this is what I've managed to accomplish for this average Navier Stokes. So it provided this sort of road map to solve the problem. Now this is a pipe dream because there are so many things that are missing for this to actually be a reality. So I I I can't create these basic logic gates.
我没有这些特殊的水配置。我的意思是，有一些候选方案包括可能可行的涡环，但模拟计算相比数字计算确实很麻烦。因为总是存在误差，你必须沿途进行大量纠错。我不知道如何完全关闭大型机器，以免干扰小型机器的运行。
I I don't I don't have these these special configurations of water. I mean, there's candidates that include vortex rings that might possibly work, but but also, you know, analog computing is really nasty compared to digital computing. I mean, because there's always errors. You you have to 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 that it doesn't interfere with the the the running of a smaller machine.
但原则上一切都有可能发生。这并不违反任何物理定律。所以这某种程度上证明了这件事是可能的。现在有其他团队正在寻找让纳维-斯托克斯爆炸的方法，这些方法远没有这个复杂得离谱。他们实际上更接近直接的自相似模型，虽然目前还不完全可行，但可能有一些比我刚才描述的更简单的方案来实现这个目标。
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 nebiospheres blow up, which are nowhere near as ridiculously complicated as this. They they actually are pursuing much closer to the the direct self similar model, which can it it doesn't quite work as is, but there could be some simpler scheme than what I just described to make this work.
这里确实有一个天才的飞跃，从纳维-斯托克斯方程到这个图灵机。所以是从什么？你试图获得越来越小的自相似团块场景。嗯。到现在有一个液体图灵机变得越来越小，然后看到这如何能用来说明爆炸问题。我的意思是，这是一个巨大的飞跃。
There is a real leap of genius here to go from Navier Stokes to this Turing machine. So it goes from what? The self similar blob scenario that you're trying to get the smaller and smaller blob Mhmm. To now having a liquid Turing machine gets smaller and smaller and smaller, and somehow seeing how that could be used to say something about a blowup. I mean, that's a big leap.
所以是有先例的。数学真的很擅长发现那些你可能认为是完全不同的问题之间的联系。但如果数学形式相同，你就可以建立联系。之前有很多我称之为细胞自动机的研究。其中最著名的是康威的生命游戏。
So there's precedent. I mean, so the the thing about mathematics that it's it's really good at spotting connections between what you think of what you might think of as completely different problems. But if if the mathematical form is the same, you you you can you can draw a connection. So there's a lot of previously on what I call cellular automata. The most famous of which is Conway's game of life.
有一个无限离散网格，在任何给定时间，网格要么被细胞占据，要么是空的。有一个非常简单的规则告诉你这些细胞如何演化。所以有时细胞存活，有时它们会死亡。你知道，当我还是个学生的时候，这实际上是一个非常流行的屏保，就是播放这些动画，它们看起来非常混乱。事实上，它们有时看起来有点像湍流。
There's this infinite discrete grid, and at any given time, the grid is either occupied by a cell or is empty. And there's a very simple rule that tells you how these cells evolve. So sometimes cells live and sometimes they they die. And this is you know, when I was a a student, it was a very popular screensaver to actually just have these these animations, like, going on, and and they they look very chaotic. In fact, they look a little bit like turbulent flow sometimes.
但在某个时候，人们在这个生命游戏中发现了越来越多有趣的结构。例如，他们发现了这个东西叫做滑翔机。滑翔机是一个非常小的配置，大约由四五个细胞组成，它会演化并朝着某个方向移动。这有点像这些涡环。是的。
But at some point, people discovered more and more interesting structures within this game of life. So for example, they discovered this thing called a glider. So a glider is a very tiny configuration of, like, four or five cells, which evolves and it just moves at a certain direction. And that's like this this vortex rings. Yeah.
所以这是一个类比。生命游戏有点像离散方程，而流体纳维-斯托克斯方程是连续方程。但在数学上，它们有一些相似的特征。随着时间的推移，人们发现了越来越多你可以在生命游戏中构建的有趣事物。生命游戏是一个非常简单的系统。
So this is an analogy. The game of life is kinda like a discrete equation, and and the fluid Navier Stokes is is a continuous equation. But mathematically, they have some similar features. And so over time, people discovered more and more interesting things that you could build within the Game of Life. The Game of Life is a very simple system.
它只有大约三四个规则，但你可以在其中设计各种有趣的配置。有一种叫做滑翔机枪的东西，它什么都不做，只是逐个吐出滑翔机。经过大量努力，人们设法为滑翔机制造了与门和或门。就像，有一个巨大而荒谬的结构，如果你有滑翔机流从这里进来，滑翔机流从这里进来，那么你可能会产生滑翔机从这里出来。如果两个流都有滑翔机，那么就会有一个输出流。
It only has, like, three or four rules to to do it, but but you can design all kinds of interesting configurations inside it. There's something called a glider gun does nothing but spit out gliders one at a one one at a time. And then after a lot of effort, people managed to create AND gates and OR gates for gliders. Like, there's this massive ridiculous structure, which if you if it if if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce extreme gliders coming out. If so may maybe if both of of the streams have gliders, then there'll be an output stream.
但如果只有一个流有滑翔机，那么什么也不会出来。嗯。所以他们可以建造类似的东西。一旦你能建造这些基本门，那么仅从软件工程的角度，你几乎可以建造任何东西。你可以建造一个图灵机。
But if only one of them does, then nothing comes out. Mhmm. So they could build something like that. And once you could build these basic gates, then just from software engineering, you can build almost anything. You can build a Turing machine.
我的意思是，这就像一个巨大的蒸汽朋克式的东西。它们看起来很荒谬。但后来人们也在生命游戏中生成了自我复制的物体。一个巨大的机器，一个骨牌机，经过很长很长的时间，并且内部总是有滑翔机枪在进行这些非常蒸汽朋克的计算。它会创建另一个版本的自己，能够复制。
I mean, the it's it's like an enormous steampunk type things. They look ridiculous. But then people also generated self replicating objects in the game of life. A massive machine, a bone over machine, which over a lot huge period of time and always glider guns inside doing these very steampunk calculations. It would create another version of itself, which could replicate
这太不可思议了。
It's so incredible.
实际上，这其中很多内容是由业余数学家群体众包的。所以我知道那项工作，这也是激励我提议对纳维-斯托克斯方程做同样事情的部分原因。正如我所说，模拟比数字差得多。你不能直接把生命游戏中的构造直接搬过来用。但这再次表明它是可能的。
A lot of this was, like, community crowdsourced by, like, amateur mathematicians, actually. So I knew about that that that work, and so that is part of what inspired me to propose the same thing with Navier Stokes. That was just a much as I said, analog is much worse than digital. Like, it's gonna be you can't just directly take the constructions in the game of life and plump them in. But, again, it just it shows it's possible.
你知道，这些细胞自动机会产生一种涌现现象。局部规则也许与流体类似。我不确定。但在大规模上运行的局部规则可以创造出这些极其复杂的动态结构。你认为这其中有什么适合数学分析的吗？
You know, there's a kinda emergence that happens with these cellular automata. Local rules maybe it's 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?
我们是否有工具能对此说出一些深刻的见解？
Do we have the tools to say something profound about that?
问题是，只有在非常精心准备的初始条件下，你才能获得这些涌现的、非常复杂的结构。是的。这些滑翔机枪、逻辑门等机器，如果你只是随机放置一些细胞，你不会看到任何这些东西。这又类似于纳维-斯托克斯的情况，你知道，在典型的初始条件下，你不会出现任何这种奇怪的计算行为。但基本上，通过工程手段，通过以非常特殊的方式专门设计，你可以选择巧妙的构造。
The thing is you can get this emergent, very complicated structures, but only with very carefully prepared initial conditions. Yeah. So so these these these glider guns and and gates and and so forth machines, if you just plunk on randomly some cells and you when you look at that, you will not see any of these. And that's the analogous situation of Navier Stokes again, you know, that that with with typical initial conditions, you will not you will not have any of this weird computation going on. But basically, through engineering, you know, by by by specially designing things in a very special way, you can pick clever constructions.
我在想是否有可能证明某种否定命题，比如基本上证明只有通过工程手段才能创造
I wonder if it's possible to prove the sort of the negative of, like, basically prove that only through engineering can you ever create
是的。是的。一些有趣的东西。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，数学中你能生成的大多数对象都是随机的。
Yeah. Yeah. Yeah. Something interesting. This 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.
它们看起来像随机的，比如圆周率的数字就是一个很好的例子。但只有极少数事物具有模式。现在你可以通过构造来证明某种模式，比如某个事物具有简单模式，你可以证明它会定期重复，你可以做到这一点。而且你可以证明，例如，大多数数字序列没有模式。
They look like random the digits of pi. Well, we believe is 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 you know, like, something has a simple pattern and you have a proof that does something like repeat itself every so often, you can do that. But and you you can prove that that for example, you can you can prove that most sequences of of digits have no pattern.
所以，就像，如果你随机选择数字，有一些很好的大数定理告诉你，从长期来看，你会得到同样多的1和2。但是，如果我给你一个特定的模式，比如圆周率的数字，我们拥有的工具就少得多，我该如何证明它没有某种奇怪的模式？我花了很多时间研究的另一项工作是证明我称之为结构定理或逆定理的东西，它们提供了测试某些事物是否具有高度结构性的方法。有些函数被称为加法函数。比如，你有一个映射自然数到自然数的函数。所以，可能，你知道，2映射到4，3映射到6，依此类推。
So, like, if if you just pick digits randomly, there's some good little large numbers that tells you you're gonna get as many ones as as twos in the long run. But we have a lot fewer tools to to 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 Some other work that I spend a lot of time on is to prove, I call, structure theorems or inverse theorems that give tests for when something is is very structured? So some functions what's called additive. Like, you have a function that matches the natural numbers, natural numbers. So so maybe, you know, two maps to four, three maps to six, and so forth.
有些函数被称为加法函数，这意味着如果你把两个输入相加，输出也会相加。例如，乘以一个常数。如果你把一个数乘以10，如果你把a加b乘以10，那等同于把a乘以10和b乘以10然后相加。所以有些函数是加法函数。有些意义是类似加法的，但不完全是加法。
Some functions are what's called additive, which means that if you add if you add two inputs together, the output gets gets added as well. For example, I multiply by a constant. If you multiply a number by 10, if you if if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and then add them together. So some function additive. Some meanings are kind of additive, but not completely additive.
举个例子，如果我取一个数n，乘以根号二，然后取它的整数部分。所以10乘以根号二大约是14点多。所以10变成了14。20变成了28。所以在这种情况下，加法性就成立了。
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 something. So 10 went up to 14. 20 went up to 28. So in that case, add additively is true then.
所以10加10是20，而14加14是28。但是因为这种取整，有时会有舍入误差。并且有时当你把a加b时，这个函数并不完全等于两个单独输出的和，而是和加减一。所以它几乎是加法的，但不完全是。数学中有很多有用的结果，我做了很多工作来发展这类理论，其效果是如果一个函数表现出这样的结构，那么基本上存在一个原因解释为什么它是真的，这个原因是因为存在另一个附近的函数，它实际上是完全结构化的，这解释了你所拥有的这种部分模式。
So 10 plus 10 is twenty, and fourteen plus 14 is 28. But because of this rounding, sometimes there's round off errors. And and sometimes when you add a plus b, this function doesn't quite give you the sum of of the two individual outputs, but the sum plus minus one. So it's almost additive, but not quite additive. So there's a lot of useful results in mathematics, and I've worked a lot in developing things like this to the effect that if if a function exhibits some structure like this, then it's basically there's a reason for why it's true, and the reason is because there's there's some other nearby function which is actually completely structured, which is explaining this sort of partial pattern that you have.
所以，在这些定理中，它创造了一种二分法，即你研究的对象要么完全没有结构，要么以某种方式与有结构的事物相关。无论哪种情况，你都可以取得进展。一个很好的例子是数学中有一个古老的定理叫做西马内利定理（Szemerédi's theorem），证明于二十世纪七十年代。它涉及尝试在一组数字中找到某种类型的模式，这些模式构成等差数列。比如3、5、7或者10、15、20。
And so if you have these so in those theorems, it it creates this dichotomy that 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 in either case, you can make progress. A good example of this is that there's this old theorem in mathematics called Simarelli's theorem, proven in the nineteen seventies. It concerns trying to find a certain type of pattern in a set of numbers that the patterns have make progression. Things like three, five, and seven or or or ten, fifteen, and 20.
而安德烈（André）已经证明，任何足够大的数字集合，称为正密度集合，都包含你希望的任何长度的等差数列。例如，奇数集合的密度是二分之一，并且它们包含任意长度的等差数列。所以在这种情况下，这很明显，因为奇数真的非常有结构。我可以直接取11、13、15、17。我可以很容易地在该集合中找到等差数列？
And I'm really Andre has already proved that any set of of numbers that are sufficiently big, called what's called positive density, has arithmetic progressions in it of of any length you wish. So for example, the odd numbers have set a density one half, and they contain arithmetic progressions of any length. So in that case, it's obvious because the the odd numbers are really, really structured. I can just take eleven, thirteen, fifteen, seventeen. I just I can I can easily find ethnic progressions in in in that set?
但是西马内利定理（Szemerédi's theorem）也适用于随机集合。如果我取奇数集合，然后抛硬币，对于每个数字，我只保留我得到正面的那些数字。因为我只是抛硬币。我只是随机去掉一半的数字。我保留一半。
But Zemetism also applies to random sets. If I take the set of odd numbers and I flip a coin and I for each number, and I only keep the numbers which for which I got a heads. Because I just flip coins. I just randomly take out half the numbers. I keep one half.
所以这是一个完全没有模式的集合。但仅仅由于随机波动，你仍然会在该集合中得到很多算术级数。你能证明这一点吗
So that's a set that has no no patterns at all. But just from random fluctuations, you will still get a lot of of ethnic progressions in that set. Can you prove that
在随机序列中存在任意长度的算术级数。是的。
there's arithmetic progressions of arbitrary length within a ran Yes.
你听说过无限猴子定理吗？通常数学家会给理论起些枯燥的名字，但偶尔他们也会起些生动的名字。是的。无限猴子定理的通俗版本是：如果你有无限数量的猴子在房间里，每只都有一台打字机，它们随机打字。几乎可以肯定，其中一只会打出整部《哈姆雷特》的剧本或任何其他有限文本字符串。
Have you heard of the infinite monkey theorem? Usually, mathematicians give boring names to theorists, but occasionally, they 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 score of Hamlet or any other finite string of text.
这只是需要一些时间，实际上需要相当长的时间。但如果你有无限数量，那么它就会发生。所以，基本上这个定理说的是，如果你取一个无限长的数字串或任何东西，最终你希望的任何有限模式都会出现。可能需要很长时间，但最终会发生。特别是，任何长度的算术级数最终都会出现。
It will just take some time, quite a lot of time, actually. But if you have an infinite number, then it happens. So, basically, the theorem says that if you take an infinite string of of digits or whatever, eventually, any finite pattern you wish will emerge. It may take a long time, but it will eventually happen. In particular, ethnic progressions of any length will eventually happen.
好的。但你需要一个极其长的随机序列才能让这种情况发生。
Okay. But you need you but you need an extremely long random sequence for this to happen.
我想这很直观。这就是无限的力量。
I suppose that's intuitive. It's just infinity.
是的。无限可以掩盖很多问题。是的。我们人类应该如何理解无限？嗯，你可以把无限看作是一个有限数字的抽象，对于这个数字你没有界限。
Yeah. Infinity absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity? Well, you can think of infinity as as as an abstraction of a finite number for which you you do not have a bound for.
你知道，我的意思是现实生活中没有什么是真正无限的。但是，你可以问这些古老的问题，比如如果我有花不完的钱会怎样？或者如果我能想跑多快就跑多快会怎样？数学家们将其形式化的方式就是，数学找到了一种形式体系来理想化处理，不是让某物变得极大或极小，而是让它真正变成无限或零。
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 if I had as much money as I wanted? You know? Or 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 or extremely small to actually be exactly infinite or zero.
通常这样做后数学会变得简洁得多。在物理学中，我们开玩笑说要假设球形奶牛。你知道，现实世界的问题有各种现实效应，但你可以理想化地将某物推向无限，将某物推向零。这样数学处理起来就变得简单多了。
And often the the mathematics becomes a lot cleaner when you do that. I mean, in physics, we we joke about assuming spherical cows. You know, like, real world problems have got all kinds of real world effects, but you can idealize send something to infinity, send something to zero. And and the mathematics becomes a lot simpler to work with there.
我想知道使用无限概念会让我们在多大程度上偏离物理现实。
I wonder how often using infinity forces us to deviate from the physics of reality.
是的。这里有很多陷阱。我们在本科数学课上花很多时间教授分析学，分析学通常就是关于如何取极限以及...比如说a加b总是等于b加a。当你有有限项时，你可以交换它们的顺序相加，这没问题。
Yeah. So there's a lot of pitfalls. So, you know, we we spend a lot of time in our undergraduate math classes teaching analysis, and analysis is often about how to take limits and and and and whether you you know? So for example, a plus b is always b plus a. So when when you have a finite number of terms, you add them, you can swap them, there's there's no problem.
但当你有无限项时，就会出现一些技巧性的游戏，你可以让一个级数收敛于某个值，但重新排列后它突然收敛到另一个值。所以你会犯错。当你允许无限时，你必须知道自己在做什么。你必须引入这些ε和δ，有一种特定的推理方式可以帮助你避免错误。近年来，人们开始研究在无限极限下成立的结果，并对它们进行理想化处理。
But when you have an infinite number of terms, they're these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and then suddenly convert it 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 and and there's there's a certain type of way of reasoning that helps you avoid mistakes. In more recent years, people have started taking results that are true in in infinite limits and and also with fantasizing them.
这是一个更定性化的问题。你可以用纯粹的有限方法来处理，运用你的有限直觉。在这种情况下，结果会随着你要生成的文本长度呈指数级增长。这就是为什么你永远看不到猴子写出《哈姆雷特》。也许能看到它们写出一个四个字母的单词，但写不出那么长的作品。
And that's a more qualitative question. And this is something that you can 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 you're trying to generate. So if and so this is why you never see the monkeys create Hamlet. You can maybe see them create a four letter word, but nothing that big.
所以我个人发现，一旦你将一个无限陈述有限化，它确实会变得直观得多，不再那么奇怪了。
And so I personally find once you finalize an infinite statement, it's it does become much more intuitive, and it's no longer so so weird.
所以即使你在处理无限性，将其有限化也是好的，这样你就能获得一些直观的理解。
So even if you're working with infinity, it's good to finitize so that you can have some intuition.
是的。缺点在于有限化的群组要混乱得多，非常混乱。是的。而且，没错。所以无限群通常先被发现，比如早几十年，然后后来人们才将它们有限化。
Yep. The downside is that the finitize groups are just much, much messier. Yeah. And and yeah. So so the infinite ones are found first, usually, like decades earlier, and then later on, people finalize them.
既然我们提到了很多数学和物理，嗯。数学和物理作为学科、作为理解和观察世界的方式，有什么区别？也许我们还可以把工程学加进来。你提到你妻子是工程师，提供了对电路的新视角。对吧。
So since we mentioned a lot of math and a lot of physics Mhmm. What is the difference between mathematics and physics as disciplines, as ways of understanding of seeing the world? Maybe we can throw an engineering in there. You mentioned your wife is an engineer, give a new perspective on circuits. Right.
所以存在一种不同的看待方式
So there's a different way of looking at
世界的方式，考虑到你从事过数学物理。所以你戴过所有的帽子，对吧。我认为科学总体上是三件事之间的互动：现实世界。
the world, given that you've done mathematical physics. So you you've you've worn all the hats. Right. So I think science in general is interaction between three things. There's the real world.
有我们对现实世界的观察，我们的观测数据，然后是我们关于世界如何运作的心理模型。我们无法直接接触现实，明白吗？我们只有观测数据，它们是不完整的，而且有误差。在很多很多情况下，我们想知道，比如明天的天气怎么样？
There's what we observe of the real world, our observations, and then our mental models as to how we think the world works. So we can't directly access reality. K? All we have are the observations, which are incomplete, and they they have errors. And there are many, many cases where we would we want to know, for example, what is the weather like tomorrow?
而我们还没有我们想要预测的那个观测数据。然后我们有这些简化模型，有时做出不现实的假设，你知道，比如球形牛那种东西。那些就是数学模型。嗯。数学关注的是这些模型。
And we don't yet have the observation that we'd like to predict. And then we have these simplified models, sometimes making unrealistic assumptions, you know, the spherical cow type things. Those are the mathematical models. Mhmm. Mathematics is concerned with the models.
科学收集观察结果，并提出可能解释这些观察结果的模型。数学的作用是我们在模型内部探讨该模型的推论是什么？模型会对未来的观察或过去的观察做出什么样的预测。它是否符合观测数据？所以这绝对是一种共生关系。
Science collects the observations, and it proposes the models that might explain these observations. What mathematics does is we we stay within the model and ask what are the consequences of that model? What observations would what predictions would the model make of the of future observations or past observations. Does it fit observed data? So there's definitely a symbiosis.
我想数学与其他学科的不同之处在于，我们从假设出发，比如模型的公理，然后探讨从这个模型中能得出什么结论。而几乎所有其他学科都是从结论出发。你知道，我想要做这个，我想要建一座桥，明白吗？
It's I guess mathematics is 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. In almost any other discipline, you start with the conclusions. You know, I want to do this. I want to build a bridge. You know?
我想要赚钱，我想要做这个。然后你寻找实现目标的路径。这其中很少有那种推测性的思考。
I I want to to make money. I want to do this. Okay? And then you you you find the path to get there. A lot there's there's a lot less sort of speculation about it.
假设我做了这个，会发生什么？你知道，就是规划和建模。也许科幻小说是另一个领域，但实际上也就这些了。我们生活中做的大多数事情都是结论驱动的，包括物理学和其他科学。
Suppose I did this. What would happen? You know, planning and and and modeling. Speculative fiction maybe is is one other place, but that's about it, actually. Most of things we do in life is conclusions driven, including physics and science.
我的意思是，他们想知道，这颗小行星会飞向哪里？明天的天气会怎样？但数学还有另一个方向，即从公理出发进行推导。
I mean, they want to know, you know, where is this asteroid gonna go? You know? What what what what is the weather gonna be tomorrow? But mathematics also has this other direction of of going from the the axioms.
你怎么看？物理学中存在着理论与实验之间的这种张力。
What do you think? There is this tension in physics between theory and experiment.
嗯。
Mhmm.
你认为什么是发现关于现实的真正新颖理念的更强大方式？
What do you think is a more powerful way of discovering truly novel ideas about reality?
嗯，你需要自上而下和自下而上两种方式。是的。只是这些事物之间确实存在互动。所以随着时间的推移，观察、理论和建模都应该更接近现实。但最初并不是这样——我的意思是，这一直都是如此。
Well, you need both top down and bottom up. Yeah. It's just that it's it's a really interaction between all these things. So over time, the observations and the theory and the modeling should go both get get closer to reality. But initially and it isn't I mean, this is this is always the case out there.
它们一开始总是相距甚远。但你需要一个来弄清楚该在哪里推动另一个。明白吗？所以如果你的模型预测了实验没有发现的异常，那就会告诉实验者去哪里寻找，你知道，去获取更多数据来完善模型。是的。
They're always far apart to begin with. But you need one to figure out where where to push the other. You know? So if your model is predicting anomalies that are not picked up by experiment, that tells experimenters where to look, you know, to to to to to find more data to refine the models. Yeah.
所以这是一个来回往复的过程。在数学本身内部，也存在理论和实验的成分。只是直到最近，理论几乎完全占据了主导地位。比如，99%的数学是理论数学，只有非常少量的实验数学。我的意思是，人们确实在做。
So it it it goes it goes back and forth. Within mathematic itself, there's there's also a theory and experimental component. It's just that until very recently, theory has dominated almost completely. Like, 99% of mathematics is theoretical mathematics, and there's a very tiny amount of experimental mathematics. I mean, people do do it.
你知道吗？比如，如果他们想研究素数或其他什么，他们可以生成大型数据集——一旦有了计算机，我们就能稍微做一点。不过即使在此之前，嗯，比如高斯，他猜想并发现了数论中最基本的定理，称为素数定理，它预测了到一百万、到一万亿有多少个素数。这不是一个显而易见的问题。基本上，他所做的是计算——我是指大部分靠自己，但也雇用了人工计算机，那些职业就是做算术的人，来计算前10万个素数之类的，制作表格并做出预测。
You know? Like, if they wanna study prime numbers or whatever, they can just generate large datasets and with a so once we had the computers, we'd be able to do it a little bit. Although even before, well, like, Gauss, for example, he discovered he conjectured the most basic theorem in in number theory to call the prime number theorem, which predicts how many primes that up to a million, up to a trillion. It's not an obvious question. And, basically, what he did was that he computed, I I mean, mostly by himself, but also hired human computers, people who whose professional job it was to do arithmetic, to compute the first 100,000 primes or something and made tables and made a prediction.
那是实验数学的一个早期例子。但直到最近，它并不……是的。我的意思是，理论数学只是更加成功。因为进行复杂的数学计算直到最近才变得可行。即使现在，你知道，尽管我们有强大的计算机，也只有一些数学问题可以通过数值方式探索。
That was an early example of experimental mathematics. But until very recently, it was not yeah. Mean, theoretical mathematics was just much more successful. I mean, because doing complicated mathematical computations is was just not not feasible until very recently. And even nowadays, you know, even though we have powerful computers, only some mathematical things can be explored numerically.
有一种叫做组合爆炸的现象。如果你想研究，比如说，定理，你想研究数字1到1000的所有可能子集。只有1000个数字。能有多糟糕呢？结果发现，1到1000的不同子集的数量是2的1000次方，这比任何计算机目前或将来能够枚举的数量都要大得多。
There's some called the combinatorial explosion. If you want us to study, for example, theorem, you wanna study all possible subsets of the numbers one to a thousand. There's only 1,000 numbers. How bad could it be? It turns out the number of different subsets of of one to a thousand is two to the power 1,000, which is way bigger than than that any computer can currently can can any computer ever or ever can enumerate.
所以，有些数学问题很快变得无法通过直接暴力计算来解决。国际象棋是另一个著名例子。我们无法让计算机完全探索所有棋局位置。但现在我们有了AI。我们有工具来探索这个空间，虽然不是100%保证成功，但可以通过实验进行。
So you have you have to be 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. But now we have AI. We have tools to explore this space not with 100% guarantees of success, but with experiment.
你知道吗？比如，我们现在可以通过经验方法解决国际象棋问题。我们有非常强大的AI，它们不需要探索游戏树中的每一个位置，但已经找到了很好的近似解法。人们实际上正在使用这些象棋引擎进行实验性象棋研究，重新审视旧的象棋理论，比如某种开局是好是坏。
You know? So, like, we can empirically solve chess now. For example, we have we have a very, very good AIs that that can you know, 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 make to do experimental chess that they're they're revisiting old chess theories about, oh, you know, when you this type of opening, this is a good this is a good type of move. This is not.
他们可以用这些象棋引擎来完善，有时甚至推翻关于象棋的传统智慧。我确实希望数学在未来能有更多的实验成分，也许由AI驱动。我们当然会讨论
And they can use these chess engines to actually refine, in some cases, overturn commercial wisdom about chess. And I do hope that that mathematics will will have a larger experimental component in the future, perhaps powered by AI. We'll, of course, talk
讨论这个问题。但在国际象棋案例中，数学也有类似情况，我不认为它提供了对不同局势的形式化解释。不，它只是说明哪个位置更好，人类可以凭直觉理解。然后我们人类可以据此构建
talk about that. But 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. No. It's just saying which position is better or not, and that you can intuit as a human being. And then from that, we humans can construct Yes.
关于这个问题的理论。你提到了柏拉图的洞穴寓言。
A theory of the matter. You've mentioned Plato's cave allegory.
嗯。
Mhmm.
如果有人不知道这个寓言：人们观察的是现实的影子而非现实本身，却相信自己观察到的就是现实。这是否在某种意义上就是数学家和所有人类正在做的事——观察现实的影子？我们是否可能真正触及现实？
So in case people don't 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? Do is it possible for us to truly access reality?
嗯，存在这三种本体论意义上的事物。有客观现实，有我们的观察，还有我们的模型。从技术上讲，它们是不同的，我认为它们将永远是不同的。没错。但它们可以随着时间的推移而越来越接近。
Well, there are these three ontological things. There's actual reality, there's our observations, and our our models. And, technically, they are distinct, and I think they will always be distinct. Right. But they can get closer over time.
你明白吗？所以，这个趋近的过程通常意味着你必须抛弃最初的直觉。就像，天文学提供了很好的例子，你知道，比如，最初的世界模型是平的，因为它看起来是平的，你知道，而且它很大，你知道，而宇宙的其余部分，天空并不是……比如太阳，看起来就非常小。所以你从一个实际上离现实很远的模型开始，但它某种程度上符合你所拥有的观察结果。你明白吗？
You know? So and the process of getting closer often means that you're you have to discard your initial intuitions. So, like, astronomy provides great examples, you know, like, you know, like, you know, initial model of the world is is flat because it it looks flat, you know, and and that it's and it's big, you know, and the rest of the universe, the sky is is not you know, like, sun, for example, looks really tiny. And so you start off with a model, which is actually really far from reality, but it fits kind of the observations that you have. You know?
所以，你知道，事情看起来不错，但是随着时间的推移，当你进行越来越多的观察，使其更接近现实时，模型也随之被拖拽着前进。因此随着时间的推移，我们不得不认识到地球是圆的，它在自转。它绕着整个系统运行。太阳系在星系中运行，如此等等。而且宇宙似乎在膨胀。
So, you know, so things look good, you know, but but over time, as you make more and more observations, bringing it closer to 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. It goes around this whole system. Solar system goes on the galaxy and so on and so forth. And the universe seems expanding.
膨胀是自我膨胀，是加速的。事实上，就在最近今年，我看到甚至有证据表明宇宙本身的加速度也不是恒定的。
Expansions are self expanding, accelerating. And in fact, very recently, in this year, I saw this even the acceleration of the universe itself is this evidence that is is nonconstant.
这背后的解释是……它正在追赶。它正在追赶。我的意思是，它仍然是，你知道，暗物质、暗能量。是的。
And explanation behind why that is It's catching up. It's catching up. I mean, it's still, you know, the dark matter, dark energy Yes.
就是这类事情。是的。我们有一个模型，可以某种程度上解释，并且非常符合数据。它只是有几个你需要指定的参数。但是，你知道，人们会说，哦，那是凑数的因子。
This kind of thing. Yes. We have a 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, you know, people say, oh, that's fudge factors.
你知道，只要有足够多的凑数因子，什么都能解释。但你必须……从数学角度看模型，你希望模型中的参数数量少于你观测数据集中的数据点数量。所以，如果你有一个包含10个参数的模型来解释10个观测结果，那是一个完全无用的模型。这被称为过拟合。但是，就像，如果你有一个，你知道，包含两个参数的模型，它解释了一万亿个观测结果，这基本上就……是的，那个暗物质模型，我认为它有大约14个参数，它解释了天文学家拥有的 petabytes 级别的数据。
You know, with with enough fudge factors, can explain anything. You have to but the mathematical point over 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 10 observations, that is a completely useless model. It's what's called overfitted. But, like, if you have a model with with, you know, two parameters and it explains a trillion observations, which is basically so, yeah, the the the dark matter model, I think it has, like, 14 parameters, and it explains petabytes of data that that that that the astronomers have.
你可以把理论想象成一种对宇宙的数据压缩。物理数学理论本质上就是一种数据压缩。你知道，我们有这些PB级别的观测数据，我们希望把它压缩成一个能用五页纸描述、指定一定数量参数的模型。如果它能以合理精度拟合几乎所有的观测数据，那么压缩得越多，你的理论就越好。
You can think of of a theory like, one way to think about physical mathematical theory theory is it's it's a compression of of the universe and a data compression. So, you know, 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 if it can fit to reasonable accuracy, you know, almost all of observations. I mean, the more compression that you make, the better your theory.
事实上，我们宇宙及其一切存在最令人惊讶之处在于它竟然是可以被压缩的。这就是数学不可思议的有效性。
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 effectiveness of mathematics.
是的。爱因斯坦有句类似的名言：宇宙最不可压缩之处在于它是可理解的。
Yeah. Einstein had a quote like that. The the most incompressible thing about the universe is that it is comprehensible.
没错。而且不仅仅是可理解。你还能写出像E=mc²这样的方程。
Right. And not just comprehensible. You can do an equation like e equals m c squared.
这其实有一些数学上的可能解释。数学中有个现象叫普适性。许多宏观复杂系统都源于大量微观相互作用。由于常见的爆炸形式，你可能会认为宏观方程必须比微观方程复杂指数倍。如果你想要完全精确求解，确实如此。
There is actually some mathematical possible explanation for that. So there's this phenomenon in mathematics called universality. So many complex systems at the macro scale are coming out of lots of tiny interactions at at the macro scale. And normally because of the common form of explosion, you would think that the macro scale equations must be, like, infinitely, exponentially more complicated than than the the macro scale ones. And they are if you want to solve them completely exactly.
比如你想模拟一盒空气中的所有原子，阿伏伽德罗常数是巨大的。粒子数量极其庞大，如果要逐个追踪将非常荒谬。但在微观尺度会涌现出某些几乎不依赖于宏观尺度、或只依赖于少量参数的规律。
Like, if you want to model all the atoms in a box of of air, that's like, Avogadro's number is humongous. Right? There's a huge number of particles. If you actually have to track each one, it'll be ridiculous. But certain laws emerge at the microscopic scale that almost don't depend on what's going on at the macro scale or or only depend on a very similar number of parameters.
所以如果你想模拟盒子里数万亿粒子组成的气体，只需要知道温度、压力、体积等五六个参数，就能模拟这些10^23个粒子几乎所有需要了解的特性。虽然我们对普适性的数学理解还远不如预期，但在一些更简单的玩具模型中，我们确实很好地理解了普适性为何出现。最基本的就是中心极限定理，它解释了为什么钟形曲线在自然界无处不在。许多事物都遵循所谓的高斯分布（著名的钟形曲线），现在甚至还有关于这个曲线的梗图。
So if you wanna model a gas of, you know, frontillion particles in box, you just need to know its temperature and pressure and volume and a few parameters, like five or six, and it models almost everything you need need to know about these 10 to 23 or whatever particles. So we we have we we don't understand universality anywhere near as we would like mathematically, but there are much simpler toy models where we do have a good understanding of why universality occurs. Most basic one is is the central limit theorem that explains why the bell curve shows up everywhere in nature. But so many things are distributed by what's called a Gaussian distribution, famous bell curve. There's now even a meme with this curve.
甚至这个迷因也具有广泛的适用性。这个迷因具有普遍性。是的。
And even the meme applies broadly. There's universality to the meme. Yes.
如果你愿意，可以上升到元层面来看，但确实存在很多很多的过程。例如，你可以取独立随机变量的损失，并以各种方式对它们进行平均。你可以取简单平均或更复杂的平均，我们可以在各种情况下证明这些钟形曲线、这些高斯分布会出现，这是一个令人满意的解释。有时它们不会出现。所以如果你有很多不同的输入，并且它们都以某种系统性的方式相互关联，那么你可能会得到与曲线相差甚远的结果。
You can go meta if you like, but there are many, many processes. For example, you you can take loss of independent random variables and average them together in in various ways. You take a simple average or more complicated average, and we can prove in various cases that that these these bell curves, these Gaussians emerge, and it is a satisfying explanation. Sometimes they don't. So so if you have many different inputs and they're all correlated in some systemic way, then you can get something very far from a curve show up.
了解这个系统何时失效也很重要。所以普遍性并不是一个100%可靠的东西。全球金融危机就是一个著名的例子。人们认为抵押贷款违约具有这种高斯类型的行为，如果你询问一个由10万美国抵押贷款持有人组成的群体，他们中有多大比例会违约。如果所有因素都是不相关的，它可能是一个钟形曲线，这样你就可以管理期权和衍生品等的风险。
And this is also important to know when this system fails. So universality is not a 100% reliable thing to rely on. That that the global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of Gaussian type behavior that that if you if you ask if a population of of of, you know, 100,000 Americans with mortgages, that's what what proportion of them will default in their mortgages. If everything was decorrelated, it could be an s bell curve, and and, like, you you can you can manage risk of options and derivatives and so forth.
这是一个非常漂亮的理论。但如果经济中出现系统性冲击，可能同时导致所有人违约，这就是非常非高斯的行为。这一点在2008年2月并未被充分考虑。现在我认为人们更意识到这是一种系统性风险。实际上这是一个更大的问题。
And and it is a very beautiful theory. But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non Gaussian behavior. And this wasn't fully accounted for in the 02/2008. Now I think there's some more awareness that this is a systemic risk. It's actually a much bigger issue.
仅仅因为模型漂亮好看，它可能并不符合现实。所以研究模型行为的数学非常重要，但同样重要的是验证模型何时符合现实、何时不符合。我的意思是，两者都需要。数学可以提供帮助，因为这些中心极限定理告诉你，如果你有某些公理，比如非相关性，如果所有输入都相互关联，那么你就会看到这些经典行为，一切都没问题。它告诉你在模型中寻找弱点的地方。
And just because the model is pretty and nice, it may not match reality. So so the mathematics of working out what models do is really important, but also the 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 it can for example, these central limit theorems, it it tells you that that if you have certain axioms like like like a non correlation, that if all the inputs are correlated to each other, then you have these classical behavior, so things are fine. It it tells you where to look for weaknesses in the model.
所以如果你对中心极限定理有数学理解，而有人提议使用这些高斯分布定律或其他什么来模拟违约风险，如果你受过数学训练，你会说：好的。但是你所有输入之间的系统性相关性是什么？然后你可以询问经济学家，这种风险有多大？然后你可以去寻找答案。所以科学和数学之间总是存在这种协同作用。
So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian copy laws or whatever to to model default risk, if you're mathematically trained, you would say, okay. But what are the systemic correlation between all your inputs? And so then the then you can ask the economists, you know, how how how much of a risk is that? And then you can you can you can go look for that. So there's always this this this synergy between science and and mathematics.
稍微谈一下普遍性这个话题。
A little bit on the topic of universality.
嗯。
Mhmm.
您以跨越数学领域令人难以置信的广度而闻名并备受赞誉，让人想起一个世纪前的希尔伯特。事实上，伟大的菲尔兹奖得主数学家蒂姆·高尔斯曾说过，您是我们最接近希尔伯特的存在。
You're known and celebrated for working across an incredible breadth of mathematics reminiscent of Hilbert a century ago. In fact, the great Fields Medal winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert.
他是您的同事。哦，是的。好朋友。
He's a colleague of yours. Oh, yeah. Good friend.
但无论如何，您以在数学上既能深入又能广博的能力而闻名。所以您是提出这个问题的完美人选：您认为是否存在连接数学所有不同领域的线索？数学整体是否具有某种深层的底层结构？
But anyway, so you 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 area areas of mathematics? Is there a kind of deep underlying structure to all of mathematics? There's
肯定存在许多连接线索，数学的许多进展可以通过以下故事来体现：选取两个先前没有联系的数学领域，并发现它们之间的联系。一个古老的例子是几何和数论。您知道吗？在古希腊时代，这些被认为是不同的学科。我的意思是，数学家们两者都研究。
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. You know? So so in the times of the ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both.
您知道，欧几里得最著名的是研究几何，但也研究数字。但它们当时并不被认为真正相关。我的意思是，有点像，您可以说这个长度是那个长度的五倍，因为您可以取这个长度的五个副本等等。但直到笛卡尔真正意识到这一点，他发展了我称之为解析几何的方法，您可以用两个实数来参数化平面这个几何对象。每个点都可以，因此几何问题可以转化为关于数字的问题。
You know, Euclid worked both on on geometry most famously, but also on numbers. But they were not really considered related. I mean, a little bit like, you know, you you could say that that that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until Descartes who really realized that he developed what I call analytic geometry that you can you can parameterize the plane, a geometric object, by by two real numbers. Every point can be and so geometric problems can be turned into into problems about numbers.
而在今天，这感觉几乎是微不足道的。就像，这其中没有什么实质内容。当然，您知道，平面就是x和y，因为这就是我们教授的内容，并且已经内化了。但这是一个重要的发展，这两个领域被统一了。这个过程在整个数学中一遍又一遍地持续进行。
And the the today, this feels almost trivial. Like, the fact that there's there's there's no content to this. Like, of course, you you know, a plane is x x and y, and because that's what we teach, and it's internalized. But it was an important development that these these two fields are are were unified. And this process has just gone on throughout mathematics over and over again.
代数和几何曾经是分开的，现在我们有了这种将它们连接起来的几何学，并且一次又一次地出现。这无疑是我最喜欢的那种数学。所以我认为数学家有不同的风格，就像刺猬和狐狸。狐狸知道很多事情但都略知一二，而刺猬则对一件事了解得非常非常透彻。
Algebra and geometry were separated, now we have this geometry that connects them and and over and over again. 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 a hedgehog knows one thing very, very well.
在数学领域，确实既有刺猬也有狐狸，还有一些人可以扮演两种角色。我认为数学家之间理想的合作需要多样性。比如狐狸与多个刺猬合作，或者反过来。所以是的，但我主要还是认同自己是一只狐狸。
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 ideal collaboration between mathematicians involves a very you need some diversity. Like, fox working with many hedgehogs or or vice versa. So yeah. But but I identify mostly as a fox, certainly.
我不知为何喜欢套利，你知道，就像学习一个领域如何运作，掌握其中的技巧，然后进入另一个人们认为不相关的领域，但我可以适应这些技巧？从而看到它们之间的联系
I I I like arbitrage somehow, you know, like 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 I can adapt the tricks? So see the connections
不同领域之间的。
between the fields.
是的。所以还有其他数学家比我深入得多。他们真的是刺猬，对一个领域了如指掌，在那个领域里他们更快、更高效。但我可以给他们这些额外的工具？
Yeah. So there are other mathematicians who are far deeper than I am. Like, who they're they're really hedgehogs. Know everything about one field, and they're they're much faster and and and more effective in that field. But I can I can give them these extra tools?
我的意思是，你说过根据情境，根据合作对象，你可以既是刺猬又是狐狸。那么，如果可能的话，你能谈谈这两种思考问题方式的区别吗？比如当你遇到一个新问题时，寻找联系与非常专注单一方向之间的区别。
I mean, you've said that you can be both the hedgehog and and the fox depending on the context Yeah. 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.
我对狐狸的模式要适应得多。是的。所以我喜欢寻找类比和叙事。如果我看到一个领域的结果并且喜欢它，我会花很多时间
I'm much more comfortable with with the the fox paradigm. Yeah. So yeah. I I like looking for analogies, narratives. I I spend a lot of time if if there's a result, I see it in one field, and I like the result.
这是个很酷的结果，但我不喜欢这个证明方式。它用到了我不太熟悉的数学工具。我经常尝试用自己偏好的工具重新证明它。通常我的证明会更差一些。但通过这个练习，我可以说，哦，现在我明白那个证明想要做什么了。
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 reprove it myself using the tools that I favor. Often, my proof is worse. But by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do.
通过这种方式，我能对那个领域使用的工具有所理解。所以这非常具有探索性，像是在陌生领域做疯狂的事情，而且，是的，很多时候是在重新发明轮子。而刺猬式的方法，我认为更加学术化。你知道，你是以知识为基础的。
And from that, I can get some understanding of of the tools that are used in in that field. So it's very exploratory, very doing crazy things in crazy fields and and, like, yeah, reinventing the wheel a lot. Yeah. Whereas, so the hedgehog style is, I think, much more scholarly. You know, you you you're very knowledge based.
你会紧跟这个领域的所有发展动态。你了解所有的历史。你对每种特定技术的优缺点都有很好的理解。是的，我认为你会更多地依赖计算，而不是试图寻找叙事。
You you you you stay up to speed on, like, all the developments in this field. You you know all the history. You have a very good understanding of of exactly the strengths and weaknesses of of each particular technique. Yeah. I think you you'd rely a lot more on sort of calculation than sort of trying to find narratives.
所以是的。我的意思是，我也可以那样做，但还有其他人在那方面非常出色。
So yeah. I mean, I could do that too, but there are other people who are extremely good at that.
让我们退一步，也许看看数学的一个有点浪漫化的版本。
Let's step back and maybe look at the the a bit of a romanticized version of mathematics.
嗯。
Mhmm.
所以我想你说过，在你生命的早期，数学更像是一种解谜活动，当你嗯。年轻的时候。你第一次遇到什么问题或证明时，意识到数学可以有一种优雅和美感？
So I think you've said that early on in your life, math was more like a puzzle solving activity when you were Mhmm. Young. When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it?
这是个好问题。当我来到普林斯顿读研究生时，约翰·康威当时就在那里。他几年前去世了。但我记得我参加的最早的研究讲座之一就是康威关于他所谓的'极端证明'的演讲。康威有一种惊人的思考方式，能以你通常不会想到的方式来思考各种问题。
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. But I remember one of the very first research talks I I went to was a talk by Conway on what he called extreme proof. So Conway just had this this amazing way of of thinking about all kinds of things in a in a way that you wouldn't normally think of.
他认为证明本身占据着某种空间。你知道吗？所以如果你想证明某个命题，比如说存在无限多个质数。你有各种不同的证明方法，但你可以沿着不同的轴线对它们进行排序。
So he thought of proofs themselves as occupying some sort of space. You know? So so if you want to prove something, let let's say that there's infinitely many primes. Okay? You have all different proofs, but you could you could rank them in different axes.
比如，有些证明很优雅，有些证明很长，有些证明很基础等等。所以这个证明云——所有证明的空间本身就有某种形状。因此他对这个形状的极端点很感兴趣。
Like, proofs are elegant. Some proofs are long. Some proofs are are elementary and so forth. And so this is cloud so the space of all proofs itself has some sort of shape. And so he was interested in in extreme points of this shape.
比如，在所有这些证明中，哪个是在牺牲其他所有方面的情况下最短的？或者是最基础的？或者其他什么？他给出了一些著名定理的例子，然后给出他认为在这些不同方面是极端证明的版本。我觉得这真的让我大开眼界——不仅仅是为结果找到一个证明很有趣，而且一旦你有了证明，尝试以各种方式优化它，证明本身也包含某种工艺性。这确实影响了我的写作风格，比如当你做数学作业、本科生的家庭作业等等时，你某种程度上被鼓励只要写下任何能成立的证明就行，然后交上去。
Like, out of all all these proofs, what is one of the the shortest at the at the excess of every everything else or or the most elementary or or whatever? And so he gave some examples of well known theorems, and then he would give what he thought was was the extreme proof in these different aspects. I I just found that really eye opening that that, you know, it's it's it's not just getting a proof for a result was interesting, but but once you have that proof, you know, trying to to to optimize it in various ways, that that proof proofing itself had some craftsmanship to it. It certainly informed my writing style that, you know, like, when you do your your math assignments and as a undergraduate, your homework, and so forth, you you're sort of encouraged to just write down any proof that works. Okay, and hand it in.
只要得到一个勾号，你就继续前进。但如果你希望你的成果真正有影响力并被人们阅读，它不能仅仅是正确的。它还应该读起来令人愉悦，有动机，能够适应推广到其他事物。这在许多其他学科中也是一样的，比如编程。数学和编程之间有很多类比。
They get a get as long as it gets a tick mark, you you move on. But if you want your your 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, you know, motivated, be adaptable to to generalized other things. It's it's the same in in many other disciplines, like like coding. It's an there's a there's a lot of analogies between math and coding.
如果你没注意到的话，我喜欢类比。但你知道，就像你可以编写一些意大利面条式的代码来完成某个任务，它快速又粗糙，但能工作。但有很多编写良好代码的好原则，这样其他人可以使用它、在此基础上构建等等，而且它bug更少等等。嗯。数学也有类似的情况。
Like I analogies if you haven't noticed. But, you know, like, 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 for writing code well so that other people can use it, build upon it, and so on, and it has fewer bugs and whatever. Mhmm. And there's there's similar things with mathematic mathematics.
所以
So
是的。首先，那里有太多美妙的事物，而康威是数学和计算机科学史上最伟大的思想家之一。仅仅是思考证明的空间就很有意思。然后说，好吧，这个空间是什么样的，它的极限在哪里？就像你提到的，编码是一个类比。
Yeah. The first of all, there's so many beautiful things there, and and Conway is one of the great minds in mathematics ever and computer science. Just even considering the space of proofs Yeah. And saying, okay, what does this space look like and what are the extremes? Like you mentioned, coding is an analogy.
有趣的是，还有一项叫做代码高尔夫的活动
It's interesting because there's also this activity called code golf
哦，是的。是的。
Oh, yeah. Yeah.
是的。我也觉得这个活动既美妙又有趣，人们使用不同的编程语言尝试写出能完成特定任务的最短程序。是的。我相信甚至还有这方面的竞赛。
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. Yeah. Then I believe there's even competitions on this.
是的。是的。是的。
Yeah. Yeah. Yeah.
是的。这也是一个很好的压力测试方式，不仅仅是测试程序，或者在这种情况下测试证明，还包括测试不同的语言。也许使用不同的符号或任何东西来完成不同的任务。
Yeah. And it's also a nice way to stress test 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 to accomplish a different task.
是的。你能学到很多东西。我的意思是，这可能看起来像是一个轻率的练习，但它能产生所有这些见解，如果你没有这个人为的目标去追求，你可能
Yeah. You learn a lot. I mean, it it may seem like a frivolous exercise, but it it can generate all these insights, which if you didn't have this artificial objective to to to pursue, you
你可能没注意到。你认为数学中最美丽或最优雅的方程是什么？我的意思是，人们通常从简洁性中寻找美。所以如果你看E=mc²。嗯。
you might not see. What do use the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in in beauty is the simplicity. So if you look at e equals m c squared. Mhmm.
所以当几个概念结合在一起时，这就是为什么欧拉恒等式常被认为是数学中最美的方程。
So when when a few concepts come together, that's why the Euler identity is often considered the most beautiful equation in mathematics.
嗯。嗯。
Mhmm. Mhmm.
你觉得欧拉恒等式美吗？
Do you do you find beauty in that one, in the Euler identity?
是的。嗯，如我所说，我觉得最吸引人的是不同事物之间的关联。比如e的iπ次方等于-1。所以，人们会说，哦，这用到了所有基本常数。好吧。
Yeah. Well, as I said, I mean, what I find most appealing is is connections between different things that you like. So the if you e to pi I equals minus one. So, yeah, people are like, oh, this uses all the fundamental constants. Okay.
那——我的意思是，那很巧妙。但对我来说，指数函数最初是欧拉用来衡量指数增长的。明白吗？所以我认为复利或衰减，或者任何持续增长、持续减少的增长和衰减，或者扩张或收缩，都是用指数函数建模的。而π来自圆和旋转。
That that that's I mean, that's cute. But but to me so the exponential function was just by Euler to measure exponential growth. Know? So I think 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 pi comes around from circles and rotation.
对吧？例如，如果你想旋转一根针100多度，你需要旋转π弧度。而复数，代表了我们想象中的旋转轴，比如90度旋转，所以是方向的变化。指数函数代表了你当前方向上的增长和衰减。当你把i放进指数时，它就不再是沿当前方向的运动，而是与当前位置成直角的运动，也就是旋转。
Right? If you want to rotate a needle, for example, 100 and degrees, you need to rotate by pi radians. And I, complex numbers, represents this swapping machine we would imagine axis of a 90 degree rotation, so a change in direction. So the exponential function represents growth and decay in the direction that you really are. When you stick an I in the exponential, it it now it's it's instead of motion in the same direction as your current position, it's the motion has a right angle to your current position, so rotation.
然后，e^(iπ) = -1 告诉你，如果你旋转π时间，最终会到达相反方向。所以它通过复数化、除以i的操作，统一了通过膨胀的几何学和指数增长动力学。这样就把所有这些数学领域连接在一起了。是的。这就是我在研究复数、复数以及复数。
And then so e e to pi equals minus one tells you that if you rotate for a time pipe, you end up at the other direction. So it unifies geometry through dilation and exponential growth dynamics through this act of of complexification, partition by by by eye. So it it connects together all these towards mathematics. Yeah. That I'm just doing complex and complex and the complex numbers.
它们几乎被认为是...是的。由于这个恒等式，它们在数学中都是近邻。
They're considered almost yeah. They're all next door neighbors in mathematics because of this identity.
你觉得你提到的事情可爱吗？这些不同领域的符号碰撞只是一个轻率的副作用，还是你认为当所有我们的老朋友符号在夜晚聚在一起时，确实存在真正的价值？
Do do you think the thing you mentioned is cute? The the the the collisional notations from these disparate fields is just a frivolous side effect, or do you think there is legitimate, like, value in when the notation all the our old friends come together night?
嗯，这这这确认了你拥有正确的概念。当你初次研究任何事物时，你必须测量事物并给它们命名。最初，有时候，因为你的模型偏离现实太远，你会给错误的事物起最好的名字，只有后来才发现什么是真正重要的。
Well, it's it's it's confirmation that you have the right concepts. So when you first study anything, you you have to measure things and give them names. And, initially, sometimes, you're because your your model is getting too far off from reality, you give the wrong things the best names, and you only find out later what's what's really important.
物理学家有时会这样做。我的意思是，但结果还好。
Physicists can do this sometimes. I mean, but it turns out okay.
所以实际上，物理学的F=ma。好吧。其中一个重要的事情就是e。对吧？当亚里士多德首次提出他的运动定律，然后是伽利略和牛顿等人，你知道，他们看到了他们能够测量的东西。
So actually, physics so equals m t squared. Okay. So one of the the big things was the e. Right? So when when Aristotle first came up with his laws of of motion and then and then Galileo and Newton and so forth, you know, they saw the things they could they could measure.
他们能够测量质量、加速度、力等等。例如牛顿力学，我认为F=ma是著名的牛顿第二运动定律。所以这些是主要对象。他们在理论中给了它们核心地位。直到后来人们开始分析这些方程时，似乎总是存在这些守恒量。
They could measure mass and acceleration and force and so forth. So Newtonian mechanics, for example, I think was MA was the famous Newton's second law of motion. So those were the the primary objects. So they gave them the central billing in the theory. It was only later after people started analyzing these equations that there always seem to be these quantities that were conserved.
特别是动量和能量。能量并不是显而易见的物理量，不像质量和速度那样可以直接测量。但随着时间的推移，人们意识到这实际上是一个极其基本的概念。19世纪，哈密顿最终将牛顿物理定律重新表述为所谓的哈密顿力学，其中能量（现在称为哈密顿量）成为了主导性的物理量。
So in particular, momentum and energy. And it's not obvious that things happen in energy. Like, it's not something you can directly measure the same way you can measure mass and and and velocity. So both but over time, people realize that this was actually a really fundamental concept. Hamilton, eventually, in nineteenth century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object.
一旦你知道如何测量任何系统的哈密顿量，你就能完全描述其动力学，即所有状态的变化情况。它确实成为了核心要素，尽管最初并不明显。这种视角转变实际上在量子力学出现时发挥了重要作用，因为早期研究量子力学的物理学家很难适应他们的牛顿思维方式——毕竟一切都被视为粒子等等。而量子力学中存在波的概念，看起来非常非常奇怪。比如有人问：量子版本的F=ma是什么？这个问题确实很难回答。
Once you know how to measure the Hamiltonian of any system, you can describe completely that the dynamics, like what what happens to to all to all the states. Like, it it 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 their Newtonian thinking because, you know, everything was a particle and so forth to to to quantum mechanics, you know, because I I think because there was a wave, it just looked really, really weird. Like, he asked, what is the quantum vision f equals MA? And it's really, really hard to to give an answer to that.
但事实证明，在经典力学中默默发挥作用的哈密顿量，也是量子力学中的关键对象——量子力学中也有一个称为哈密顿量的对象。虽然它是不同类型的对象（称为算符而非函数），但同样地，一旦确定了它，你就确定了整个动力学。存在一个方程，只要有了哈密顿量，就能精确告诉你量子系统如何演化。
But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in in quantum mechanics that this there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than than a function. But and but, again, once you specify it, you specify the entire dynamics. So there's equation that tells you exactly how quantum systems evolve once you have a Hamiltonian.
并列来看，它们看起来是完全不同的对象。一个涉及粒子，一个涉及波等等。但通过这种中心性，你实际上可以开始将许多直觉和事实从经典力学转移到量子力学。例如，在经典力学中有个叫做诺特定理的东西。
So side by side, they look completely different objects. You know? Like, one involves particles, one involves waves, and so forth. But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So example, in in classical mechanics, there's this thing called Noether's theorem.
每当物理系统存在对称性时，就存在一个守恒定律。物理定律具有平移不变性——如果我向左移动10步，我所经历的物理定律与在原地时相同，这对应着动量守恒。如果我旋转某个角度，同样经历相同的物理定律，这对应着角动量守恒。
Every time there's a symmetry in a physical system, there is a conservation law. So the laws of physics are translation invariant. Like, if I move 10 steps to left, I experience the same laws of physics as if I was here, and that corresponds to conservation moment momentum. If I turn around by by some angle, again, I experience the same laws of physics. This corresponds to conservation of angular momentum.
如果我等待十分钟，物理定律仍然相同。所以存在时间平移不变性，这对应着能量守恒定律。因此对称性与守恒性之间存在这种基本联系，这在量子力学中同样成立，尽管方程完全不同。但因为它们都源自哈密顿量，而哈密顿量控制着一切。
If I wait for ten minutes, I still have the same laws of physics. So there's time transition variance. This corresponds to the lower conservation of 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. But because they're both coming from a Hamiltonian, Hamiltonian controls everything.
每当哈密顿量具有对称性时，方程就会存在守恒定律。所以一旦掌握了正确的语言，事情就会变得清晰很多。我们至今无法统一量子力学和广义相对论的原因之一，就是我们还没有弄清楚基本对象是什么。例如，我们必须放弃时空是近似欧几里得型空间的概念，而且我们知道在极微小尺度上会出现量子涨落，存在时空泡沫。
Every time the Hamiltonian has a symmetry, the equations will will have a conservation law. So it's it's it's it's once you have the right language, it actually makes things a lot a lot cleaner. One of problems why we can't unify quantum mechanics and general relativity yet, haven't we figured out what the fundamental objects are. Like, for example, we have to give up the notion of space and time being these almost Euclidean type spaces, and it has to be you know, and, you know, we kind of know that at very tiny scales, there's gonna be quantum fluctuations. There's a space space time foam.
试图使用笛卡尔坐标x y z将会是行不通的。但我们不知道用什么来替代它。我们实际上缺乏相应的数学概念——类似于哈密顿量但能组织一切的类比物。
And trying to to use Cartesian coordinates x y z is gonna be it's it's just it's it's a nonstarter. But we don't know how to what to replace it with. We don't actually have the mathematical concepts. The analog of the Hamiltonian, but sort of organized everything.
你的直觉认为存在万物理论吗？也就是说，找到统一广义相对论和量子力学的语言是可能的吗？
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?
我相信是的。物理学的历史就是一部统一史，就像数学一样。电和磁曾经是独立的理论，后来麦克斯韦统一了它们。牛顿统一了天体运动与地球物体的运动等等。所以统一应该会发生。
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 were were separate theories, then Maxwell unified them. You know, Newton unified the the motion of the heavens for the motions on of objects on the earth and so forth. So it should happen.
问题在于，回到观测与理论的模型——物理学是其自身成功的受害者。我们两大物理理论，广义相对论和量子力学，现在都极其精确。它们共同覆盖了99.9%的可观测现象。你必须通过极端强大的粒子加速器、早期宇宙或难以测量的领域，才能发现与这两个理论的偏差，从而找到结合它们的方法。但我相信，几个世纪以来我们一直在进步，没有理由会停止。
It's just that the again, to go back to this model of the of the observations and and and theory, part of our problem is that physics is a victim of its own success, that our two big theories of of of physics, general relativity, and quantum mechanics are so are so good now. 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 accelerations or or the early universe or or or things that are really hard to measure in order to get any deviation from either of these two theories to a point where you can actually figure out how to how to combine them together. But I have faith that we, you know, we've we've been doing this for centuries. We've made progress before, and there's no reason why we should stop.
你认为自己会成为发展万物理论的数学家吗？
Do you think you will be a mathematician that develops theory of everything?
通常当物理学家需要某种数学理论时，数学家往往早已提前研究过相关雏形。当爱因斯坦意识到空间是弯曲的时，他去找数学家询问是否已有西方学者提出过有用的弯曲空间理论。对方回答：是的，黎曼提出过相关理论。果然，黎曼发展的黎曼几何几乎正是爱因斯坦理论所需的空间理论。
What often happens is that when the physicists need some theory of mathematics, there's often some precursor that the mathematicians worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, you know, is there is there some theory of curved space that the westerners already came up with that could be useful? And he said, oh, yeah. There's I think of Riemann came up with something. And so, yeah, Riemann had developed Riemannian geometry, which is precisely, you know, a theory of spaces that occurred in in various general ways, which turned out to be almost exactly what was needed by Einstein's theory.
这回到了维格纳提出的'数学在自然科学中不可思议的有效性'。能很好解释宇宙的理论，往往也涉及能很好解决数学问题的数学对象。希望它们只是组织数据的两种不同方式
This is going back to to Wiemann's unreasonable effectiveness of mathematics. I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical problems. Hopefully, they're just sort both ways of organizing data
以各种有用的方式。只是感觉你可能需要去某个难以直观理解的奇怪领域。就像，是的，是的。比如弦理论。
in in in in useful ways. It just feels like you might need to go some weird land that's very hard to to intuit. Like Yeah. Yeah. You have, like, string theory.
是的。
Yeah.
那个理论是几十年来一直是主要候选理论。我认为它正在慢慢失宠，因为它与实验不符。
That that's that was that was a leading candidate for many decades. It's I think it's slowly falling out of fashion because it's it's not matching experiment.
所以，正如你所说，一个重大挑战当然是实验非常困难。是的。因为这两种理论的有效性都很高。但另一个挑战是，你不仅在偏离时空概念，还要进入一些疯狂的多维空间。
So one of the big challenges, of course, like you said, is experiment is very tough Yes. Because of the how effective Yeah. 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.
是的。你在做各种奇怪的事情，对我们来说，我们已经从最初提到的平坦地球概念走得太远了。
Yeah. You're doing all kinds of weird stuff that to us, we've gone so far from this flat earth that we started at, like you mentioned.
是的。是的。
Yeah. Yeah.
是的。是的。现在我们很难用我们作为猿类后代的有限认知能力去直观理解那个现实到底是什么样子。
Yeah. Yeah. Now we're just it's it's very hard to use our limited ape descendants of a a cognition to intuit what that reality really is like.
这就是为什么类比如此重要。你知道吗？我的意思是，确实如此。地球是圆的并不直观，因为我们被困在它上面。但是，你知道，但一般来说，我们对圆形物体有着相当不错的直觉。
This is why analogies are so important. You know? I mean, so yeah. The round earth is not intuitive because we're we're stuck on it. But, you know, but you you you know, but round objects in general, have pretty good intuition over.
而且我们一直对光的工作原理等等很感兴趣。实际上，这是一个很好的练习，去弄清楚日食、月相等等如何能够通过圆形地球和圆形月亮的模型轻松解释。你可以拿一个篮球、一个高尔夫球和一个光源，自己动手演示这些现象。所以直觉是存在的，但确实需要迁移应用。
And we've been interested about light works and so forth. And, like, it's it's actually a good exercise to actually work out how eclipses and phases of of the sun and the moon and so forth can be really easily explained by by by by round earth and round moon, you know, and models. And and you can just take, you know, a basketball and a golf ball and and and a light source and actually do these things yourself. So the intuition is there, but, yeah, you have to transfer it.
对我们来说，从扁平地球到圆形地球在智力上是一个巨大的飞跃
That is a big leap intellectually for us to go from flat to round earth
嗯。
Mhmm.
因为，你知道，我们的生活大多是在平面世界里度过的。是的。要加载这些信息，我们都，像是，认为理所当然。我们对太多事情都视为理所当然，因为科学已经为这类事情建立了大量证据。但，你知道，我们其实是住在一块圆形的岩石上。是的。
Because, you know, our life is mostly lived in flatland. Yeah. To load that information, and we're all, like, 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.
在太空中飞行。是的。是的。这是一个巨大的飞跃，而且随着我们不断进步，你必须进行一连串这样的飞跃，越来越多。
Flying through space. Yeah. Yeah. And it's a big leap, and you have to take a chain of those leaps the more and more and more we progress.
没错。是的。所以现代科学也许，再次地，成为了自身成功的受害者，也就是说，为了更加精确，它不得不离你最初的直觉越来越远。因此，对于一个没有经历过完整科学教育过程的人来说，它看起来越来越可疑。是的，就是因为这个原因。
Right. Yeah. So modern science is maybe, again, a victim of its own success is that, you know, 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 and more suspicious Yeah. Because of that.
所以，你知道，我们需要更多的接地气。我是说，我认为，你知道，有些科学家做了很出色的科普工作，但有很多科学实验你可以在家里完成。有很多YouTube视频。我最近做了一个和之前提到的格兰特·桑德森合作的视频，展示了古希腊人是如何测量月球距离、地球距离的，使用的技术你自己也能复现。不一定都需要像高级太空望远镜那样高大上，或者非常复杂的数学。
So, you know, we we we need we need more grounding. I mean, I I think I mean, you know, there are there are scientists who do excellent outreach, but there's there's there's there's there's lots of science things that you can do at home. I I there's lots of YouTube videos. I did a YouTube video recently of Grant Sanderson that we talked about earlier that, you know, how the ancient Greeks were able to measure things like the distance to the moon, distance to the earth, and, you know, using techniques that you you could also replicate yourself. It doesn't all have to be like fancy space telescopes and and very intimidating mathematics.
是的。我强烈推荐那个。我记得你做过一次讲座，还和格兰特合作了一个非常精彩的视频。尝试将自己代入那个时代人的思维，是一种美妙的体验
Yeah. That's I highly recommend that. I believe you gave 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
嗯。
Mhmm.
笼罩在神秘之中。对吧。你知道，就像你在这个星球上，不知道它的形状、大小，看到一些星星，看到一些东西，然后尝试在这个世界中定位自己。是的。是的。并尝试对到某些地方的距离做出一些概括性的陈述。
Shrouded in mystery. Right. 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.
视角的转变非常重要。你看旅行能开阔视野。这是智力上的旅行，你知道。把自己代入古希腊人或其它时代的人的思维中。做出假设，比如球形牛之类的，什么都行。
Change of perspective is really important. You see travel broadens the mind. This is intellectual travel, you know. Put yourself in the mind of the ancient Greeks or or some other person from other time period. Make hypotheses, spherical cows, whatever.
嗯。进行推测。你知道，这就是数学家所做的，某种程度上艺术家其实也这样做。
Mhmm. Speculate. And, you know, this is this is what mathematicians do and some sort of artists do, actually.
令人难以置信的是，在极端限制下，你仍然能说出非常有力量的话。这就是为什么它如此鼓舞人心。回顾历史，即使拥有的不多，也能弄清楚很多事情。对吧。当你资源有限时。没错。
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 Right. When you don't have much. Right.
图
Figure
弄清楚东西。如果你提出公理，那么数学允许你，你遵循这些公理得出它们的结论，有时候你可以从初始假设出发走得很远很远。
out stuff. If you propose axioms, then the mathematics lets you, you follow those axioms to to their conclusions, and sometimes you can get quite a lot quite a long way from, you know, initial hypotheses.
如果我们停留在奇妙的领域，你提到了广义相对论。你对爱因斯坦场方程的数学理解做出了贡献。你能解释一下这项工作吗？从数学的角度来看，广义相对论的哪些方面让你觉得有趣，对你具有挑战性？
If we stay in the land of the weird, you mentioned general relativity. You've 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?
我研究过一些方程。有一个叫做波映照方程或西格玛场模型的东西，它不完全是时空引力本身的方程，而是可能存在于时空之上的某些场的方程。所以爱因斯坦的相对论方程只是描述空间和时间本身。但除此之外还有其他的场存在，比如电磁场。
I have worked on some equations. There's something called the the wave maps equation or 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. So Einstein's equation of relativity just describes space and time itself. But then there's other fields that live on top of that. There's the elect electromagnetic field.
还有叫做杨-米尔斯场的东西。这一整套不同方程的层次结构中，爱因斯坦方程被认为是最非线性和最困难的之一。但在这个层次结构中相对较低的是这个叫做波映照方程的东西。所以它是一个波，在任何给定点都固定在一个球面上。我可以想象在时空中有一堆箭头，这些箭头指向不同的方向，但它们像波一样传播。
There's things called Yang Mills fields. And there's this whole hierarchy of different equations, of which Einstein is considered one of the most nonlinear and difficult. But relatively low on the 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. So I can think of a bunch of arrows in space and time and and and and the arrow is pointing in in different directions, but they propagate like waves.
如果你摆动一个箭头，它会传播并使所有的箭头移动，有点像麦田里的麦浪。我再次对这个问题的全局正则性问题感兴趣。就像，是否可能让所有的能量都聚集在一个点上。所以我考虑的方程实际上被称为临界方程，它在所有尺度上的行为大致相同。我勉强能够证明，你实际上无法迫使所有能量集中在一个点的情景发生，能量必须稍微分散一点。目前来说，只要稍微分散一点点，它就能保持正则性。
If if if you wiggle an arrow, it'll it'll propagate and create and make all the arrows move kinda like sheets of wheat in the wheat field. And I was interested in the global regularity problem again for this question. Like, is it possible for for all the energy here to to to to collect at a So the equation I considered was actually was 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 that you couldn't actually force a scenario where all the energy concentrated at one point, that the energy had to disperse a little bit. At the moment, just a little little bit, it it would it would stay regular.
是的。这是二月份的事了。这实际上也是我后来对纳维-斯托克斯方程产生兴趣的部分原因。是的。所以我开发了一些技术来解决那个问题。
Yeah. This was back in February. That was part of why I got interested in Nary Soaks afterwards, actually. Yeah. So I developed some techniques to solve that problem.
所以部分原因是这个问题由于球面曲率而非常非线性。存在某种非线性效应，这是一种非微扰效应。当你正常看待它时，它看起来比波动方程的线性效应更大。因此，即使能量很小，也很难控制局面。但我发展出了一种称为规范变换的方法。
So part of it is it was this problem is really nonlinear because of the curvature of the sphere. There's there was a certain nonlinear effect, which was a nonperturbative effect. It was when you sort of looked at it normally, it it looked larger than the linear effects of the wave equation. And so it was hard to to keep things under control even when the energy was small. But I developed what's called a gauge transformation.
所以这个方程有点像麦浪的演化，它们都在来回弯曲。因此有很多运动。但就像，如果你想象通过在空间不同点附加小摄像头来稳定流动，这些摄像头试图以捕捉大部分运动的方式移动。在这种稳定流动下，流动变得更加线性。我发现了一种变换方程的方法来减少非线性效应的量，然后我能够解出这个方程。
So the equation is kinda like an evolution of of of heaves of wheat, and and they're all bending back and forth. And so there's a lot of motion. But, like, if you imagine, like, stabilizing the flow by attaching little cameras at different points in space, which are trying to move in a way that captures most of the motion. And under this sort stabilized flow, the flow becomes a lot more linear. I discovered a way to transform the the equation to reduce the amount of nonlinear effects, and then I was able to to to to solve the equation.
我在澳大利亚参观我的艺术作品时发现了这个变换，当时我试图理解所有这些场的动力学，我无法用纸笔计算。也没有足够的计算机来进行任何模拟。所以我最终闭上眼睛，躺在地板上，想象自己实际上就是那个场，四处滚动，试图找到一种改变坐标的方式，使得各个方向的事物都能以相对线性的方式行为。是的，我姑姑在我这样做的时候走了进来，她问我，你在做什么？答案很复杂。
I found this transformation while visiting my art in Australia, and I was trying to understand the dynamics of all these fields, and I I couldn't do a pen and paper. And I had not enough facilitative computers to do any computer simulations. So I ended up closing my eyes, being on on the floor, just imagining myself to actually be the specter field and rolling around to try to to see how to change coordinates in such a way that somehow things in all directions would behave in a reasonably linear fashion. And, yeah, my aunt walked in on on me while I was doing that, and she was asking, what are that what am I doing doing this? It's complicated is the answer.
是的。你知道，她说，好吧。好吧。你知道，你是个年轻人。我不多问。
Yeah. And, you know, she said, okay. Fine. You know, you're a young man. I don't ask questions.
我我必须问一下，你知道，你如何解决难题？如果可能进入你的思维，当你在思考时，你是在脑海中可视化数学对象、符号吗？你通常思考时在脑海中可视化什么？
I I I have to ask about the you know, how do you approach solving difficult problems? What if it's possible to go inside your mind when you're thinking, are you visualizing in your mind the mathematical objects, symbols maybe? What are you visualizing in your mind usually when you're thinking?
很多纸笔工作。作为数学家，你学到的一点是我称之为战略性作弊。数学的美在于你可以随心所欲地改变问题和规则。这在其他任何领域都做不到。比如，如果你是工程师，有人说要在这条河上建一座桥，你不能说我想把桥建在那里，或者我想用纸而不是钢来建造。
A lot of pen and paper. One thing you pick up as a mathematician is sort of I call it cheating strategically. So the the the beauty of mathematics is that is that you get to change the rule change the problem and change the rules as you wish. This you don't get to do this for any other field. Like, you know, if if you're an engineer and someone says, put a bridge over this this river, can't say, I wanna build this bridge over here instead, or I wanna put it out of paper instead of steel.
但数学家，你可以做任何你想做的事。这就像试图解决一个电脑游戏，你有无限的作弊码可用。所以，你知道，你可以设置这个，如果维度太大，我就把它设为一。我会先解决一维问题。
But a mathematician, you can you can do whatever you want. It's it's like trying to solve a computer game where you can there's unlimited cheat codes available. And so, you know, you you can you can set this so there's a dimension that's too large. I'll set it to one. I'd solve the one dimensional problem first.
所以有一个主项和一个误差项。我要做一个球形牛假设，假设误差项为零。解决这些问题的方法不是那种钢铁侠模式，把一切都弄得极其困难。实际上，处理任何合理数学问题的方式应该是：如果有10个因素让你头疼，就找一个版本的问题，关掉其中九个困难，只保留一个。这样你就相当于安装了九个作弊码。
So there's a main term and an error term. I'm gonna make a spherical cow assumption as I assume the error term is zero. And so the way you solve these problems is is not in sort of this iron man mode where you make things maximally difficult. But, actually, the way you should you should approach any reasonable math problem is that you if if there if there are 10 things that are making your life difficult, find a version of problem that turns off nine of the difficulties but only keeps one of them. And so that and then that just so you you you you install nine cheats.
好的。如果你安装10个作弊码，游戏就变得琐碎了。但你只安装九个作弊码。你解决一个问题，这个问题教会你如何处理那个特定的困难。然后你关掉这个，再打开另一个，解决那个问题。
Okay. If you install 10 cheats, then then the game is trivial. But you install nine cheats. You solve one problem that that that that teaches you how how to deal with that particular difficulty. And then you turn that one off, and you turn someone else something else else on, and then you solve that one.
当你学会分别解决这10个问题10个困难后，你就需要开始一次合并几个。我小时候看过很多香港动作电影。这是一种文化。有一点是，每次打斗场面，你知道，某些电影里主角被100个坏蛋喽啰围攻之类的。但编排总是让你一次只和一个人打，打败那个人后再继续前进。
And after you you know how to solve the 10 problems 10 difficulties separately, then you have to start merging them a few at a time. I I as a kid, I watched a lot of these Hong Kong action movies. It's from a culture. And one thing is that every time it's a fight scene, you know, some movie that the hero gets swarmed by a 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 it would defeat that person and move on.
正因为如此，他才能打败所有人。对吧？但如果他们打得更聪明一点，一拥而上同时围攻那个人，就会变成糟糕得多的电影编排，那样他们就会赢了。
And and because of that, he could he could defeat all of them. Right? But whereas if they had fought a bit more intelligently and just swarmed the guy at once, it would make for much much worse chore cinema that that they would win.
你通常用纸笔工作吗？还是用电脑和LaTeX？
Are you usually pen and paper? Are you working with computer and latex?
我主要是用纸笔，实际上。我的办公室里有四块大黑板，有时候我需要把关于问题的所有已知信息都写在整块黑板上，然后坐在沙发上，整体审视这一切。
I'm mostly pen and paper, actually. So in in my office, I have four giant blackboards, and sometimes I just have to write everything I know about the problem on the full blackboards and then sit my couch and just sort of see the whole thing.
全是符号和记号吗？还是有一些图画？
Is it all symbols like notation, or is there some drawings?
哦，有很多绘图和很多只有我自己才懂的定制涂鸦。我的意思是，这就是黑板的美妙之处——你可以擦掉它，这是一件非常有机的事情。我开始越来越多地使用电脑，部分原因是人工智能让简单的编码变得容易多了。但你知道，如果以前我想绘制一个函数，这个函数有点复杂，需要一些迭代之类的，我得记住如何设置Python程序，循环怎么工作，然后调试它，这可能需要两个小时等等。而现在我可以在十到十五分钟内完成。
Oh, there's lot of drawing and a lot of bespoke doodles that that only make sense to me. I mean and and that's the beauty of blackboards you erase, and it's it's it's a it's a very organic thing. I'm beginning to use more and more computers partly because AI makes it much easier to do simple coding things. But, you know, if I wanted to plot a function before, which is moderately complicated, has some iteration or something, you know, I'd had to to remember how to set up a Python program and and and and and and how does a full loop work and and and debug it, and it would take two hours and so forth. And and now I can do it in ten, fifteen minutes.
是的，确实如此。我越来越多地使用电脑进行简单的探索。
It's it's much yeah. I'm I'm using more and more computers to do simple explorations.
如果可能的话，我们来稍微谈谈人工智能。也许一个好的切入点是先谈谈计算机辅助证明。你能描述一下Lean形式化证明编程语言吗？它如何作为证明助手提供帮助，也许还可以谈谈你如何开始使用它以及它对你有什么帮助。
Let's talk about AI a 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 and how it can help as a proof assistant and maybe how you started using it and how it has helped you.
Lean是一种计算机语言，很像标准的语言如Python和C等。不同之处在于，大多数语言的焦点是生成可执行代码。代码行会执行操作，比如翻转比特位、让机器人移动或在互联网上传递文本等。Lean也是一种可以做到这些的语言。
So Lean is a computer language much like sort of standard languages like Python and C and and so forth. Except that in most languages, the focus is on producing executable code. Lines of code do things. You know, they they flip bits or or they make a robot move or or they they deliver you text on the Internet or something. So Lean is a language that can also do that.
它也可以作为标准的传统语言运行，但它还能生成证书。像Python这样的软件语言可能会执行计算并告诉你答案是7。好吧，它计算3加4等于7。但Lean不仅能给出答案，还能提供证明，说明它是如何得到3加4等于7的答案，以及所有涉及的步骤。所以它创建的是更复杂的对象，不仅仅是陈述，而是带有证明的陈述。
It can also be run as a standard traditional language, but it can also produce certificates. So a software language like Python might do a computation and give you that 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 but a proof that how it got the the answer of seven as three plus four and all the steps involved in in in in so it's a so it creates these more complicated objects, not just statements, but statements with proofs attached to them.
每一行代码都是一种将先前的陈述拼接在一起以创建新陈述的方式。这个想法并不新鲜。这些东西被称为证明助手，它们提供了可以创建相当复杂、精细的数学证明的语言。它们生成这些证书，如果你信任Lean的编译器，就能100%保证你的论证是正确的。但他们让编译器变得非常小，而且有几种不同的编译器可供选择。
And every line of code is just a way of piecing together previous statements to to create new ones. So the idea is not new. These things are are called proof assistance, and so they provide languages for which you you can create quite complicated, intricate mathematical proofs. And they produce these certificates that give a 100% guarantee that your arguments are correct if you trust the compiler of Lean. But they made the compiler really small, and you can there are several different compilers available for the same for
你能给大家一些直观的感受，说明用笔和纸写作与使用Lean编程语言之间的区别吗？形式化一个正确的陈述有多难？
Can you give people some intuition about the the difference between writing on pen and paper versus using lean programming language? How hard is it to formalize Right. Statement?
所以很多数学家都参与了Lean的设计。它的设计理念是让单行代码类似于单行逻辑论证。比如，你可能想要引入一个变量，想要证明一个矛盾。你可以做各种标准操作，它的编写方式理想情况下应该是一一对应的。
So lean a lot of mathematicians were involved in the design of lean. So it's it's designed so that individual lines of code resemble individual lines of ethical argument. Like, you might want to introduce a variable. You wanna wanna prove a contradiction. You you're there are various standard things that you can do, and and it it's it's written so ideally, it should be like a one to one correspondence.
实际上并非如此，因为Lean就像是在向一个极其迂腐的同事解释证明，他会指出：你真的确定是这个意思吗？如果这个是零会怎样？你如何证明这个？
In practice, it isn't because Lean is, like, explaining a proof to an extremely pedantic colleague who will will point out, okay. Did you really mean this? Like, what what happens if this is zero? Okay. Did you how do you justify this?
所以Lean内置了很多自动化功能来减少烦琐操作。例如，每个数学对象都必须有类型。如果我提到x，x是实数、自然数、函数还是什么？非正式写作时这些都在上下文里。你会说：显然x等于y和z的和，而y和z已经是实数，所以x也应该是实数。
So Lean has a lot of automation in it to try to to to be less annoying. So for example, every mathematical object has to come with a type. Like, if I if I talk about x, is x a real number or a natural number or or or a function or something? If you write things informally, it's it's up in the term context. You say, you know, clearly, x is equal to let x be the sum of y and z, and y and z were already real numbers, so x should also be a real number.
Lean能处理大部分这种情况，但偶尔它会说：等等，你能告诉我更多关于这个对象的信息吗？它是什么类型的对象？明白吗？你需要从更哲学的角度思考，不仅是你正在进行的计算，还包括每个对象在某种意义上的本质。
So Lean can do a lot of that, but every so often, it it says, wait a minute. Can you tell me more about what this object is? What what type of object it is? You see? You have to think more at a philosophical level about not just sort of computations that you're doing, but sort of what each object actually is in some sense.
他是在使用类似LLM的技术进行类型推断吗？就像你刚才提到的实数例子？
Is he using something like LLMs to do the type inference or, like, you mentioned with the real number?
它使用的是更传统的技术，所谓的老式人工智能。你可以把所有这些东西表示为树结构，总有算法能将一棵树与另一棵树匹配。
It's it's using much more traditional, what's called good old fashioned AI. You can represent all these things as trees, and there's always algorithm to match one tree to another tree.
所以实际上判断某个东西是实数还是自然数是可行的？
So it's actually doable to figure out if something is a a real number or a natural number?
是的。每个对象类型都带有其来源的历史记录，你可以追溯它。哦，我明白了。是的。所以它是为可靠性而设计的。
Yeah. Every object sort comes with a history of where it came from, and you can you can kinda trace it. Oh, I see. Yeah. So it's it's it's designed for reliability.
所以现代AI并没有被使用，它是一种分离的技术。人们开始基于Lean使用AI。当数学家尝试在Lean中编写证明时，通常会有一步。好的。现在我想使用微积分基本定理，比如说，来做下一步。
So modern AIs are not used in it's a disjoint technology. People are beginning to use AIs 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 theorem of calculus, say, okay, to do the next step.
所以Lean开发者构建了这个名为Mathlib的大型项目，这是一个包含数万个数学对象有用事实的集合。其中某处就有微积分基本定理，但你需要找到它。所以现在的很多瓶颈实际上是引理搜索。你知道，有一个工具就在那里某个地方，你需要找到它。因此，你可以使用专门为Mathlib设计的各种搜索引擎。
So the lean developers have built this this massive project called Methylib, a a collection of tens of thousands of useful facts about mathematical objects. And somewhere in there is the fundamental theorem of calculus, but you need to find it. So a lot of bottleneck now is actually lemma search. You know, there's a tool that that you know is in there somewhere, and you need to find it. And so you can there are various search engines specialized for Methylib that you can do.
但现在有这些大型语言模型，你可以说，好的。我此时需要微积分基本定理。我说，好的。例如，当我编码时，我在IDE中安装了GitHub Copilot插件，它会扫描我的文本，看到我需要什么。它可能会说，你知道，我甚至可能会输入它。
But there's now these large language models that you can say, okay. I need the fundamental calculus at this point. And I said, okay. For example, when I code, I have GitHub Copilot installed as a plug in to my IDE, and it scans my text, it sees what I need. It says, you know, I might even type it.
好的。现在我需要使用微积分基本定理。好的。然后它可能会建议，试试这个。
Okay. Now I need to use the fundamental calculus. Okay. And then it it might suggest, okay. Try this.
而且，大约25%的时间，它完全正确。另外15%的时间，它不完全正确，但足够接近，我可以说，哦，是的。如果我就在这里和这里改一下，它就能工作。然后大约一半的时间，它给我的是完全没用的东西。所以人们开始在一定程度上使用AI，主要是在花哨的自动补全层面上，你可以输入半行证明，它会告诉你。
And, like, maybe 25% of the time, it works exactly. And then another 15% of the time, it doesn't quite work, but it it's close enough that I can say, oh, yeah. If I just change it here and here, it will work. And then, like, half the time, it gives me complete rubbish. So but people are beginning to use AIs a little bit on top, Mostly on the level of basically fancy autocomplete that you can type half of one line of a proof, and it will find it will tell you.
是的。但是
Yeah. But but
一个花哨的功能，特别是那种带大写字母F的花哨功能，就是消除数学家从纸笔转向形式化时可能感受到的一些摩擦。是的。
a fancy, especially fancy with the sort of capital letter f, is remove some of the friction Yeah. Mathematician might feel when they move from pen and paper to formalizing.
是的。没错。所以目前我估计，形式化一个证明所需的时间和精力大约是手写出来的10倍。是的。所以虽然可行，但确实很烦人。
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 to write it out. Yeah. So it's doable, but you don't it's it's annoying.
但这难道不会，像是，扼杀
But doesn't it, like, kill
数学家工作的整个氛围吗？
the whole vibe of being a mathematician?
是的。我的意思是，就像有个吹毛求疵的同事。对吧。是的。如果这是唯一的问题的话。
Yeah. So, I mean Having a pedantic coworker. Right. Yeah. If if that was the only aspect of it.
好吧。但是，实际上有些方面因为形式化反而更令人愉快。我形式化了一个定理，最终陈述中得出了一个常数12。这个12必须贯穿整个证明过程，所有内容都需要检查，确保与其他数字一致，最终得到这个数字12。然后我们写了一篇论文，里面用了12这个数字。
Okay. But, okay, there there's some there's some things because it was actually more pleasant to do this formally. So there's a there's a theorem I formalized, and there's a certain constant 12 that that came out of that in in the final statement. And so this 12 had to be carried all through the proof, And, like, everything had to be checked that it goes all the all these other numbers that had to consistent with this final number 12. And then so we wrote a paper with number 12.
几周后，有人说通过重新调整某些步骤，实际上可以把12改进为11。用纸笔工作时，每次改变参数，都必须逐行检查证明的每一行是否仍然成立。可能会有些你完全没有意识到的微妙之处，某些你甚至没意识到自己在利用的成熟性质。所以证明可能会在某个微妙的地方崩溃。
And then a few weeks later, someone said, oh, we can actually improve this 12 to an 11 by reworking some of these steps. And when this happens with pen and paper, like, every time you change a 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 properties that are mature that you didn't even realize that you were taking advantage of. So a proof can break down at a subtle place.
所以我们用常数12形式化了这个证明。然后当这篇新论文发表时，我们说，好吧。这花了大约三周时间和20个人来形式化这个原始证明。我说，哦，但现在让我们把12更新为11。在Lean中你可以这样做，只需在脑海中找到定理，把12改成11。
So we had formalized the proof with this constant 12. And then when this this new paper came out, we said, okay. Let's so that took, like, three weeks to formalize and and, like, 20 people to formalize this this this original proof. I said, oh, but now now let's let's let's update the twelve to eleven. And what you can do with Lean, so you just in your head find theorem, you've you change your 12 to 11.
你运行编译器，在你拥有的数千行代码中，90%仍然有效，只有几行显示为红色。现在我无法证明这些步骤的合理性，但它立即隔离了哪些步骤需要修改。你可以跳过所有正常工作的部分。如果你用良好的编程实践正确编写代码，大部分代码行不会变红。只有少数地方需要修改——我的意思是，如果你没有硬编码常量，而是...
You run the compiler, and, like, of the thousands of lines of code you have, 90% of them still work, and there's a couple that are lined in red. Now I can't justify these these steps, but it immediately isolates which steps you need to change. But you can skip over everything which which works just fine. And if you program things correctly with some good programming practices, most of your lines will not be red. And there'll just be a few places where you I mean, if if you don't hard code your constants, but you sort of Mhmm.
你使用智能策略等方法，可以将需要修改的内容定位到非常短的时间内。所以在一两天内，我们就更新了证明——这是一个非常快速的过程。你做一个更改，现在有10个地方不工作。对每个问题你进行修改，然后又出现五个新问题。
You use smart tactics and so forth, you can you can localize the things you need to change to to a very small period of time. So it's like within a day or two, we had updated our proof to this is a very quick process. You you make a change. There are 10 things now that don't work. For each one, you you make a change, and now there's five more things that don't work.
但这个过程比纸笔方式收敛得更加顺畅。
But but the process converges much more smoothly than with pen and paper.
这是关于写作的。你能阅读它吗？比如如果有人有另一个证明，你能...与纸质论文相比怎么样？
So that's for writing. Are you able to read it? Like, if somebody else has a proof, are you able to, like how what's what's the versus paper and
是的。证明更长，但每个独立部分更容易阅读。如果你拿一篇数学论文，跳到第27页看第六段，有一行文字或数学公式，我通常不能立即读懂，因为它假设了各种定义，我必须回溯到可能10页前查看定义。证明分散在各处，你基本上被迫按顺序阅读。不像小说那样，理论上你可以从中间打开开始阅读。
Yeah. So the proofs are 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 of text or math, I often can't read it immediately because it it assumes various definitions, which I have to to go back and and and maybe 10 pages earlier, this was defined. And this the proof is scattered all over the place, and you basically are forced to read fairly sequentially. It's it's not like, say, a novel where, like, you know, in theory, you could you open up a novel halfway through and and start reading.
需要很多上下文。但在Lean中，如果你把光标放在一行代码上，每个对象都可以悬停查看，它会显示这是什么、来自哪里、如何证明的。追踪内容比翻阅数学论文容易得多。所以Lean真正实现的是在原子级别上协作证明，这在过去是不可能的。传统上，纸笔方式下，当你想与另一位数学家合作时，要么在黑板上真正互动。
There's a lot of context. But when I put in Lean, if you put your cursor on a line of code, every single object there, you can hover over it, and it would it would say what it is, where it came from, where it's sort justified. You can trace things back much easier than sort of flipping through a 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. So traditionally, pen and paper, when you wanna collaborate with another mathematician, either you do it at a blackboard where you you can really interact.
但如果你是通过电子邮件之类的方式来做这件事，基本上，是的，是的，你必须分段处理。所以我要完成第三部分，你做第四部分，但你们无法真正同时在同一件事上协作。然而使用Lean，你可以尝试形式化证明的某个部分，然后说，我在第67行这里卡住了。我需要证明这个，但它不太行得通。
But if you're doing it sort of by email or something, basically, yeah, yeah, you have to segment it. So I'm gonna I'm gonna finish section three. You do section four, but you can't really sort of work on the same thing, collaborate at the same time. But with Lean, you can be trying to formalize some portion of proof and say, I got stuck at line 67 here. I need to prove this thing, but it it doesn't quite work.
这是我遇到困难的代码片段。但由于所有上下文都在那里，其他人可以说，哦，好的。我明白你需要做什么。你需要应用这个技巧或这个工具，这样你们可以进行极其原子级别的对话。所以多亏了Lean，我可以与世界各地数十人协作，其中大多数人我从未见过面。
Here's the, like, the freelance code I'm having 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 to to apply this trick or this tool, and you can do extremely atomic level conversations. So because of Lean, I can collaborate, you know, with dozens of people across the world, most of whom I don't have never met in person.
我甚至可能不知道他们在证明认证方面有多可靠，但Lean给了我一个信任证书。所以我可以进行无需信任的数学工作。
And I may not know actually even whether they're how reliable they are in in in their, in in the proof certificate, but Lean gives me a certificate of of trust. So I can do I can do trustless mathematics.
这里面有太多有趣的问题了。首先，你以擅长协作而闻名。那么，在协作解决数学难题时，正确的方法是什么？是采用分而治之的方式，还是头脑风暴？你们是专注于特定部分并进行 brainstorming 吗？
So there's so many interesting questions there. So one, you're 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 brains? Are you focusing on a particular part and you're brainstorming?
总是先有一个头脑风暴的过程。是的。数学研究项目本质上就是这样，当你开始时，你并不真正知道如何解决这个问题。它不像工程项目，其他理论已经确立了几十年，主要困难在于实施。你甚至需要弄清楚什么是正确的路径。
There's always a brainstorming process first. Yeah. So math research projects sort 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 it's it's implementation is the main difficulty. You have to figure out even what is the right path.
所以这就是我之前说的关于先'作弊'的意思。你知道吗？就像回到建桥的类比。首先，假设你有无限的预算和无限的劳动力等等。
So so this is what I said about about cheating first. You know? It's like to go back to the bridge building analogy. You know? So first, assume you have infinite budget and and, like, unlimited amounts of of of workforce and so forth.
现在你能建这座桥吗？好的。好的。现在有无限预算，但只有有限的劳动力。对吧？
Now can you can you build this bridge? Okay. Okay. Now have infinite budget, but only finite workforce. Right?
现在你能做到这一点吗？我的意思是，当然，没有工程师能真正做到这一点。就像我说的，他们有固定的要求。是的。一开始总会有这种即兴讨论环节，你尝试各种疯狂的想法，做出各种不切实际的假设，但计划稍后修正。
Now can you do that and so forth? So, I mean, of course, no no engineer can actually do this. Like I said, they have fixed requirements. Yes. There's this sort of jam sessions always at the beginning where you try all kinds of crazy things, and you you you make all these assumptions that are unrealistic, but you plan to fix later.
你试图看看是否至少存在某种可能可行的基本方法框架。然后希望将问题分解成更小的子问题，虽然你不知道如何解决，但你会专注于这些子问题。有时候不同的合作者更擅长处理某些特定方面。我以本·格林定理闻名，现在称为格林陶定理。该定理指出素数包含任意长度的等差数列。
And you try to see if there's even some skeleton of an approach that might work. And then hopefully, breaks up the problem into smaller subproblems, which you don't know how to do, but then you you focus on on on the sub ones. And sometimes different collaborators are better at at working on on certain things. So one of my theorems I'm known for is a theorem of Ben Green, which is now called the Green Tau theorem. It's a statement that the primes contain ethnic progressions of any event.
所以这是对该定理的一个改进。我们的合作方式是：本已经证明了长度为三的等差数列的类似结果。他展示了就像素数包含大量长度为三的等差数列那样，甚至素数的某些子集也包含这类数列。但他的技术只适用于三项等差数列，无法适用于更长的数列。而我当时掌握来自遍历理论的技术，这是我一直在研究且比本更熟悉的领域。
So it's a modification of this theorem already. And the way we collaborated was that Ben had already proven a similar result for progressions of length three. He showed that just like the primes contain loss and loss of progressions of length three, even and even subsets of the prime, certain subsets But his techniques only worked for for the three progression. They didn't work for longer progressions. But I had these techniques coming from a goddess theory, which is something that I had been playing with and and I knew better than Ben at the time.
因此，如果我能够证明某个与素数相关的集合具有特定的随机性性质——存在某个技术条件，如果本能够为我提供这个事实，我就能完成定理的证明。但我提出的其实是数论中一个非常困难的问题，他说不可能证明这个。于是他问：你能不能在你那部分证明中使用一个较弱的前提条件，让我有机会证明它？
And so if I could justify certain randomness properties of set some set relating to primes. Like, there's there's a certain technical condition, which if I could have it if if Ben could supply me to this fact, I could give I could conclude the theorem. But I what I asked was a really difficult question in number theory, which he said, no. There's no way we can prove this. Can so he said, can you prove your part of the theorem using a weaker hypothesis that I have a chance to prove it?
他提出了一个他能证明的条件，但对我来说太弱了，无法使用。于是我们就这样反复讨论。是的，这就是不同的'作弊'方式。
And he proposed something which he could prove, but it was too weak for me. I can't use this. So there's this there's this conversation going back and forth. Yeah. So so The different cheats too.
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是的。是的。我想多'作弊'一些，他想少'作弊'一些。但最终，我们找到了一个性质：a) 他能证明，b) 我能使用，然后我们就证明了我们的定理。
Yeah. Yeah. I wanna cheat more. He wants to cheat less. Mean but eventually, we found a a a property which, a, he could prove, and b, I could use, and then we we could prove our view.
是的。所以存在各种动态变化。你知道，每次合作都有其独特的故事，没有两次是完全相同的。
And yeah. So there's there's a there are all kinds of dynamics. You know? I mean, it's it's it's every every collaboration has a has a has some story. No two are the same.
而另一方面，正如你提到的，使用精益编程。嗯。这几乎是完全不同的情况，因为你可以创建，我想你提到过一种蓝图。嗯。对。针对一个问题，然后你确实可以用精益方法进行分而治之，分别处理不同的部分
And then on on the flip side of that, like you mentioned, with lean programming Mhmm. 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 Mhmm. Right. For a problem, and then you can really do a divide and conquer with lean where you're working on separate parts
没错。
Right.
他们使用计算机系统证明检查器，基本上是的。来确保整个过程一切正确。
And they're using the computer system proof checker, essentially Yeah. To make sure that everything is correct along the way.
这样就让所有东西都兼容且可信。是的。所以目前只有少数数学项目能以这种方式拆分。以当前的技术水平，大部分精益活动都是在形式化那些已经被人类证明过的定理。数学论文在某种意义上就是一个蓝图，它把一个困难命题，比如大定理，分解成100个小引理，但通常不会写得足够详细到每个都能直接形式化。
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 boosts that have already been proven by humans. A math paper basically is a boob a blueprint in a sense that it is taking a a difficult statement, like big theorem, and breaking up into a 100 little lemmas, but often not all written with enough detail that each one can be sort of directly formalized.
蓝图就像一个极其迂腐的论文版本，其中每一步都尽可能详细解释，并试图让每一步自成一体，或者只依赖于非常特定数量的先前已证明命题，这样生成的蓝图图中的每个节点都可以独立处理。你甚至不需要知道整个系统如何运作。这就像现代供应链。你知道吗？就像如果你想制造一部iPhone或其他复杂物品，没有一个人能单独造出整个产品。
A blueprint is like a really pedantically written version of a paper where every step is explained as as much detail as as as possible and to try to make each step kind of self contained and or depending on only a very specific number of of previous statements have been proven so that each node of this blueprint graph that gets generated can be tackled independently of of 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 wanna create an iPhone or or some other complicated object, no one person can can build up a single object.
但你可以有专家，如果他们从其他公司得到一些小部件，就能把它们组合起来形成一个稍大一点的部件。
But you can have specialists who who just if they're given some widgets from some other company, they can combine them together to form a slightly bigger widget.
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。对。分解的问题，那么你就可以有成千上万的贡献者。对吧？
I think that's a really exciting possibility because you can have if you can find problems that could be Right. Broken down this way, then you could have, you know, thousands of contributors. Right?
是的。是的。分布式的。我之前跟你讲过理论数学和实验数学之间的划分。现在大部分数学都是理论性的，只有极小部分是实验性的。
Yes. Yes. Distributed. So I told you before about the split between theoretical and experimental mathematics. And right now, most mathematics is theoretical, only a tiny bit is experimental.
我认为Lean以及GitHub等其他软件工具平台，它们将使得实验数学能够扩展到比我们现在能做到的更广泛的程度。目前如果你想对某些数学模式进行探索，你需要编写代码来展现这个模式。虽然有些计算机代数包可以提供帮助，但通常都是数学家自己编写大量Python代码之类的。由于编码容易出错，让别人协作编写代码模块是不现实的，因为如果一个模块有bug，整个系统就不可靠了。
I think the platform that Lean and and other software tools, so GitHub and things like that, allow they will allow experimental mathematics to be to scale up to a much greater degree than we could do now. So right now, if you want to do any mathematical exploration of some mathematical pattern or something, you need some code to 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 prone 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 is unreliable.
所以你会看到这些由非专业程序员（数学家）编写的定制化意大利面条式代码。它们笨重且运行缓慢。正因为如此，大规模产出实验结果是很困难的。不过确实如此。
So it's these are so you get these bespoke spaghetti code written by non not professional programmers, mathematicians. You know? And they're clunky and and and slow. And so because of that, it's it's it's hard to to really mass produce experimental results. But yeah.
但我认为借助Lean，我已经开始了一些项目，我们不仅是在实验数据，更是在实验证明。我有一个叫做
But I think with Lean I mean, so 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 Equation of Theories project. Basically, generated about 22,000,000 little problems in abstract algebra. Maybe I should back up and and tell you what what the project is. Okay.
抽象代数研究像乘法和加法这样的运算及其抽象性质。比如乘法具有交换律，X乘以Y总是等于Y乘以X，这是针对数字而言的。
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. These are for numbers.
它还具有结合律。X乘以Y再乘以Z等于X乘以(Y乘以Z)。所以这些运算遵守某些定律而不遵守其他定律。例如，X乘以X并不总是等于X，所以这个定律并不总是成立。
And it's also associative. X times y times z is the same as x times y times z. So these operations obey some laws that that don't obey others. For example, x times x is not always equal to x. So that law is not always true.
因此，对于任何运算，它都遵守某些定律而不遵守其他定律。我们生成了大约4000种代数可能定律，某些运算可能满足这些定律。我们的问题是，哪些定律可以推导出其他定律？例如，交换律是否意味着结合律？答案是否定的，因为确实存在遵守交换律但不遵守结合律的运算。
So given any any operation, it obey some laws and not others. And so we generated about 4,000 of these possible laws of algebra that certain operations could 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 doesn't obey the assertion of law.
所以通过构造一个例子，你可以证明交换律并不蕴含结合律。但其他一些定律确实通过替换等方式蕴含其他定律，你可以写下一些代数证明。因此我们考察这4000条定律之间的所有配对，大约有2200万对。对于每一对，我们问：这条定律是否蕴含那条定律？如果是，给出证明。
So by producing an example, you can you can show that commutativity does not imply associativity. But some other laws do imply other laws by substitution and so forth, and you can write down some some algebraic proof. So we look at all the pairs between these 4,000 laws, and there's about twenty twenty two million of these pairs. And for each pair, we ask, does this law imply this law? If so, give a give a proof.
如果不是，给出反例。嗯。所以是2200万个问题，每一个都可以交给比如本科生代数学生，他们有很大几率能解决问题。虽然至少有2200万个问题，但其中大约100个左右确实相当困难。好吧。
If not, give a counterexample. Mhmm. So 22,000,000 problems, each one of which you could give to, like, an an undergraduate algebra student, and they had a decent chance of solving the problem. Although there are a few at least 22,000,000, there are, like, a 100 or so that are really quite hard. Okay.
但很多都很简单。这个项目就是要弄清楚整个关系图，也就是哪些定律蕴含哪些其他定律。
But a lot are easy. And the project was just to to work out to determine the entire graph, like like, which ones imply which other ones.
顺便说一句，这真是个了不起的项目。如此好的想法。对我们一直在讨论的内容进行了如此好的检验，而且规模如此惊人。
That's an incredible project, by the way. Such a good idea. Such a good test of the very thing we've been talking about at a scale that's remarkable.
是的。所以这本来是不可行的。你知道吗？我的意思是，文献中的最新进展大概是15个方程以及它们如何应用的程度。这差不多是人类用纸笔能处理的极限了。
Yeah. So it it would not have been feasible. You know? I mean, the state of the art in the literature was, like, you know, 15 equations and sort of how they apply it. That's sort of at the limit of what a human pen and paper can do.
所以需要扩大规模。需要众包，但也需要信任所有...我的意思是，没有人能亲自检查这2200万个证明。对吧？需要计算机化。所以只有使用Lean才成为可能。
So so you need to scale it up. So you need to crowdsource, but you also need to trust all the you know, I I mean, no one person can check 22,000,000 of these proofs. Right? You you need to be computerized. And so it only became possible with with Lean.
我们还希望大量使用人工智能。这个项目几乎完成了。在这2200万对中，除了两对之外都已解决。哇。实际上，对于那两对，我们已经有手写的证明，正在将其形式化。
We were hoping to use a lot of AI as well. So the the project is almost complete. So of these 22,000,000, all but two have been settled. Wow. And well, actually, and of those two, we have a pen and paper proof of the two, and we we're formalizing it.
事实上，今天早上我一直在忙着完成它。所以我们差不多快完成了。这太不可思议了。是的，太棒了。
In fact, I was this morning, I was working on finishing it. So we're almost done on this. It's Incredible. It's yeah. Fantastic.
有多少人能够参与？大约50人。在数学领域，这被认为是一个巨大的数字。
How many people were able to get About 50. Which in mathematics is is considered a huge number.
这确实是个巨大的数字。太疯狂了。
It's a huge number. That's crazy.
是的。所以我们有一篇有50位作者的论文，还有一个详细的附录说明每个人的贡献。这里有个有趣的问题，也许可以更广泛地讨论一下。当你拥有这样一个人员池时，
Yeah. So we're have a paper of 50 authors and a a big appendix of who contributed what. Here's an interesting question, not to maybe speak even more generally about it. When you have this pool of people,
有没有办法根据人员的专业水平来组织贡献，所有贡献者？好吧，我在这里问了很多天马行空的问题，但我设想的是未来可能有一群人类，也许还有一些AI。是的，能不能像Elo评分系统那样，
is there a way to organize the contributions by level of expertise of the people, all the contributors? Now okay. I'm asking a lot of pothead questions here, but I I'm imagining a bunch of humans and maybe in the future some AIs Yeah. Can there be, like, an Elo rating type of situation where,
就像把这个游戏化？这些Lean项目的妙处在于，你自动就能获得所有这些数据。是的，就像这次所有的东西都上传到了GitHub，GitHub会追踪每个人的贡献。所以你可以从中生成统计数据。
like, a gamification of this? The beauty of of these Lean projects is is that automatically, you get all this data. Yeah. So, like like, everything's been uploaded for this GitHub, and GitHub tracks who contributed what. So you could generate statistics from yeah.
在任何后续时间点，你都可以说，哦，这个人贡献了这么多行代码之类的。我的意思是，这些都是非常粗略的指标。我绝对不希望这变成，比如说，你十年评审的一部分之类的。但是，我认为在企业计算领域，人们确实已经使用其中一些指标作为员工绩效评估的一部分。这又是学术界不太愿意走的一个有点吓人的方向。
At any at any later point in time, you can say, oh, this person contributed this many 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, you know, part of your ten year review or something. But, I mean, I think already in in in enterprise computing, right, people do use some of these metrics as part of of the assessment of of performance of a of an employee. Again, this is the direction which is a bit scary for academics to go down.
我们我们我们不太喜欢指标。
We we we don't like metrics so much.
但学术界确实使用指标。他们只是用老的指标。论文数量。
And yet academics use metrics. They just use old ones. Number of papers.
是的。是的。确实是这样没错。我的意思是
Yeah. Yeah. It's true it's true that yeah. I mean
感觉这个指标虽然有缺陷，但至少是朝着更正确的方向前进。对吧？是的。这很有趣。至少是个非常有趣的指标。
It feels like this is a metric while flawed is is going in the more in the right direction. Right? Yeah. It's interesting. At least it's a very interesting metric.
是的。我认为研究它很有意思。我的意思是，你可以研究这些是否是更好的预测指标。有个问题叫古德哈特定律。如果一个统计数字被实际用来激励绩效，它就会被操纵，然后就不再是一个有用的衡量标准了。
Yeah. I think it's interesting to study. I mean, I think you can do you can do studies of of of whether these are better predictors. There's this problem called Goodhart's law. If a statistic is actually used to incentivize performance, it becomes gamed, and then it is no longer a useful measure.
哦，人类总是
Oh, humans always
是的。是的。我知道。我的意思是，这这这是理性的。所以我们这个项目做的是自我报告。
Yeah. Yeah. I know. I mean, it's it's it's rational. So what we've done for this project is is self report.
实际上，科学界有关于人们所做贡献类型的标准分类。所以有概念验证、资源提供、编码等等。我们有一个标准的贡献者类别列表。我们只是要求每个贡献者在一个包含所有作者和所有类别的大矩阵中勾选他们认为自己贡献的方框，给出一个大致的概念，比如，你做了一些编码，提供了一些计算资源，但没有做任何纸笔验证之类的。我认为这样是可行的。
So there are actually standard categories from the sciences of what types of contributions people give. So there's there's a concept and validation and resources and and and and and coding and and so forth. So we we we there's a standard list of pro or so categories. And we just ask each contributor to there's a big matrix of all the of all the authors and all the categories just to tick the boxes where they think that they contributed and just give a rough idea, you know, like, also, you did some coding and and and you provided some compute, but you didn't do any of the pen and paper verification or whatever. And I think that that works out.
传统上，数学家只是按姓氏字母顺序排列作者。所以我们没有像科学界那样的传统，比如第一作者、第二作者等等，这是我们引以为豪的。我们让所有作者地位平等，但这在如此大规模的项目中不太适用。十年前，我参与了一些叫做'多数学家项目'的工作，那是众包数学，但没有包含Lean组件。
Traditionally, mathematicians just order alphabetically by surname. So we don't have this tradition as in the 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.
所以它受到限制，因为需要人工审核员来检查所有提交的贡献是否有效，这实际上是一个巨大的瓶颈。但我们仍然完成了大约10位作者参与的项目。当时我们决定不区分每个人的具体贡献，而是使用一个统一的化名。我们创造了一个名为DHJ Polymath的虚构角色，灵感来自20世纪著名数学家群体的化名传统。论文以这个化名发表，所以我们都没有获得正式的作者署名。
So it was limited by you needed a human moderator to actually check that all the contributions coming in were actually and and this was a huge bottleneck, 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. So we created this fictional character called DHJ Polymath in the spirit of is the pseudonym for a famous group of mathematicians in the twentieth century. But and so the paper was auth authored on the pseudonym, so none of us got the author credit.
事实证明这样做并不太好，有几个原因。一是如果你想要申请终身教职之类的，你不能把这篇论文作为你的出版物提交，因为你没有正式的作者署名。但我们后来意识到的另一个问题是，当人们引用这些项目时，他们自然会提到项目中最著名的人物。哦，所以这是Tim Gowers的项目，这是Terrence Tao的项目，而不提其他19位左右参与的人。
This actually turned out to be not so great for a couple of reasons. So so one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your as you submitted as a one of your publications because it was you didn't have the formal author credit. But the other thing that we've recognized much later is that when people referred to these projects, they naturally referred to the most famous person who was involved in the project. Oh, so this was Tim Gowers' project. This was Terrence Tao's project, not mention the the other 19 or whatever people that were involved.
哦，是的。
Oh, yeah.
所以这次我们尝试不同的方法，每个人都作为作者，但我们会有一个包含这个矩阵的附录，看看效果如何。
So we're trying something different this time around where we have everyone's an author, but we will have an an appendix with this matrix, and we'll see how that works.
我的意思是，这两个项目都非常了不起。你能参与如此大规模的合作本身就令人惊叹。但我想我几年前看过Kevin Buzzard关于Lean编程语言的演讲，他说这可能是数学的未来。令人兴奋的是，作为世界上最伟大的数学家之一，你正在拥抱这个看似铺就数学未来的事物。所以我必须问你关于人工智能如何整合到这个过程中的问题。
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 Buzzard about the Lean Programming Languages a few years ago, and he was 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. So I have to ask you here about the integration of AI into this whole process.
所以DeepMind的Alpha证明是通过强化学习训练的，嗯。使用了失败和成功的正式Lean证明
So DeepMind's alpha proof was trained using reinforcement learning Mhmm. On both failed and successful formal lean proofs
嗯。
Mhmm.
针对IMO问题。是的。是的。算是高水平的中学，哦，非常高水平的。是的。
Of IMO problems. Yes. Yes. Sort of high level high school Oh, very high level. Yes.
非常高水平的中学数学问题。你对这个系统有什么看法，也许这个能证明中学水平问题的系统与
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 Right.
与更高层次问题之间的差距？是的。证明涉及的步骤数量增加时，难度呈指数级增长。这是组合爆炸问题。对吧？
Versus gradual level problems? Yeah. The the difficulty increases exponentially with the the number of steps involved in the proof. It's a combinatorial explosion. Right?
所以大型语言模型的问题是它们会犯错。如果一个证明有20个步骤，而你在每个步骤上有10%的失败率，比如走错方向，那么实际上到达终点的可能性就非常小。
So the thing of large language models is is that they make mistakes. And so if a proof has got 20 steps and your has a 10 failure rate at each step of of going the wrong direction, like, it's it's just extremely unlikely to actually reach the end.
实际上，稍微偏离一下主题，从自然语言映射到正式程序的这个问题有多难？
Actually, just to take a small tangent here, is how hard is the problem of mapping from natural language to the formal program?
哦，是的。实际上这极其困难。自然语言，你知道，它非常容错。就像，你可以犯一些小的语法错误，说第二语言的人也能大致理解你在说什么。是的。
Oh, yeah. It's extremely hard, actually. Natural language, you know, it's very fault tolerant. Like, you can make a few minor grammatical errors, and the speaker in the second language can get some idea of what you're saying. Yeah.
但是但是形式语言，是的，你知道，如果如果你犯了一个小错误，我认为整个事情就就就是胡言乱语。
But but formal language, yeah, you know, if if you get one little thing wrong, I think that the whole thing is is is nonsense.
明白
Got
了。即使是形式语言到形式语言也也也非常困难。存在存在不同的不兼容的证明系统语言。有Lean，但也有Coq和Isabelle等等。实际上，即使是从一种形式语言转换到另一种形式语言也也基本上是一个未解决的问题。
it. Even formal to formal is is is very hard. There are there are different incompatible proof of system languages. There's Lean, but also Cock and Isabelle and and so forth. And, actually, even converting from a formal language to formal language is is an unsolved basically, unsolved problem.
这太迷人了。好的。但是一旦你有了非正式语言，他们使用他们的RL训练模型，类似于他们用来尝试提出boost的AlphaZero。我相信他们还有一个单独的模型用于几何问题。那么这个系统让你印象深刻的地方是什么，你认为差距在哪里？
That is fascinating. Okay. So but once you have an informal language, they're using their RL trained model, so something akin to AlphaZero that they used to go to then try to come up with boost. They also have a model, I believe it's a separate model for geometric problems. So what impresses you about this system, and what do you think is the gap?
是的。
Yeah.
我们之前谈到过，那些随着时间的推移变得惊人的事物会逐渐变得常态化。所以，是的，现在不知怎的，当然，几何问题是一个银弹问题。
We talked earlier about things that are amazing over time become kind of normalized. So, yeah, now somehow it's of course, geometry is a silver bullet problem.
对。确实如此。确实如此。我的意思是，它依然很美。
Right. That's true. That's true. I mean, it's still beautiful.
是的。是的。不。这是一项伟大的工作，展示了可能性。我的意思是，目前这种方法还不具备可扩展性。
Yeah. Yeah. No. It's it's a great work that shows what's possible. I mean, it's it the approach doesn't scale currently.
是的。一个高中数学问题就耗费了谷歌服务器三天的计算时间。这这不是一个可扩展的前景，尤其是随着复杂度的增加呈指数级增长。
Yeah. Three days of Google's server is server time due to one high school math problem there. This this is not a scalable prospect, especially with the exponential increase in as as the complexity increases.
我们应该提到他们获得了银牌级别的表现。相当于。
We should mention that they got a silver medal performance. The equivalent of.
我的意思是，是的。银牌级别的表现。首先，他们花费的时间远超规定时限，并且有人类通过形式化提供协助。但是，是的，他们也因为解决方案获得了满分，我猜是经过形式验证的。所以我认为这是公平的。
I mean Yeah. Of a silver medal performance. So first of they took way more time than was allotted, and they had this assistance where where the humans started helped by by formalizing. But, yeah, also, they're they're giving us those full marks for the solution, which I guess is formally verified. So I guess that that's that's fair.
有努力，将来某个时候会有人提议举办一个AI数学奥林匹克竞赛，在人类选手拿到题目的同时，AI也会在相同时间内解决同样的问题，并且输出将由同一批评委评判。这意味着必须用自然语言而非形式语言来书写。
There there are efforts there was there will be a proposal at some point to actually have an an AI math Olympiad where at the same time as the human contestants get the the actual little bit problems, AIs will also be given the same problems with the same time period, and the outputs will have to be created by the same judges. And which means that it will have to written in natural language rather than formal language.
哦，我希望这能实现。我希望这个IMO能举办。我希望我希望下一次
Oh, I hope that happens. I hope that this IMO happens. I hope I hope the next
第一点。我认为这不会发生。在这个时间段内，性能还不够好。而且，但有一些较小的竞赛。有些竞赛的答案是一个数字，而不是长篇证明。
one. It won't happen this IMO. The performance is not good enough in in in the time period. And and but there are smaller competitions. There are competitions where the the answer is a is a number rather than a a long form proof.
而AI实际上在那些有具体数值答案的问题上表现要好得多，因为很容易对其学习进行强化。是的。你得到了正确答案。你得到了错误答案。这是一个非常清晰的信号。
And that's that's AI is actually a lot better at problems where there's a specific numerical answer because it's it's it's easy to to to reinforce to reinforce some learning on it. Yeah. You got the right answer. You got the wrong answer. It is it's it's a very clear signal.
但长篇证明要么必须是形式化的，然后Lean可以给出通过或不通过，要么是非形式化的。但那样就需要人类来验证。嗯。如果你试图进行数十亿次的强化学习运行，你无法雇佣足够的人类来创建这些验证。我的意思是，对于之前的语言学习模型来说，仅对人们获得的常规文本进行强化学习已经够困难了。但现在如果你实际雇佣人员，不仅仅是给出通过或不通过，而是实际从数学上检查输出，是的，那太昂贵了。
But a long form proof either has to be formal, and then the lean can give a thumbs up, thumbs down, or it's informal. But then you need a human to create it Mhmm. To tell and if you're trying to do billions of of reinforcement learning, you know, runs, you're you're not you can't hire enough humans to to create those. I mean, it's already hard enough for for the last language learners to do reinforcement learning on on just the regular text that that people get. But now if you actually hire people, not just give thumbs up, thumbs down, but actually check the the output mathematically, yeah, that's too expensive.
那么如果我们只是探索这个可能的未来，人类在数学中做的最特别的事情是什么，以至于你可以看到AI在一段时间内无法攻克。是发明新理论，提出新猜想而不是证明猜想。对吧。构建新的抽象，新的表示法，也许是像图灵那样的AI风格，看到不同领域之间的新联系？这是个好问题。
So if we just explore this possible future, what what what is the thing that humans do that's most special in 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. Right. Building new abstractions, new representations, maybe an AI turner style with seeing new connections between disparate fields? That's a good question.
我认为数学家们随着时间的推移所做的事情的本质已经发生了很大变化。你知道，一千年前，数学家们必须计算复活节的日期，有非常复杂的计算，但几个世纪以来这些都自动化了。我们不再需要这些了。你知道？
I think the nature of what mathematicians do over time has changed a lot. You know? So a thousand years ago, mathematicians had to compute the date of Easter, and there's really complicated calculations, you know, but it's all automated been automated for centuries. We don't need that anymore. You know?
他们过去用球面三角学进行导航，从旧世界到新世界。所以非常复杂的计算已经被自动化了。你知道，甚至很多本科数学，在AI之前，比如Wolfram Alpha，虽然不是语言模型，但可以解决很多本科级别的数学任务。所以在计算方面，验证常规的事情，比如有一个偏微分方程的问题，你能用20种标准技术中的任何一种解决它吗？
They used to navigate to do spherical navigation spherical trigonometry to navigate how to get from from the old world to the new. So I get very complicated calculation that can be been automated. You know, even a lot of undergraduate mathematics, even before AI, like Wolfram Alpha, for example, is is not a language model, but it can solve a lot of undergraduate level math tasks. So on the computational side, verifying routine things like having a a problem and and say, here's a problem in partial differential equations. Could you solve it using any of the 20 standard techniques?
它们会回答是的。我尝试了所有20种，这里有100种不同的排列组合，以及我的结果。这类事情，我认为会运作得很好。一旦你解决了一个问题，让AI攻击100个相邻问题的这种扩展能力。人类仍然在做的事情，所以AI目前真正挣扎的是知道何时走错了方向。
And they have yes. I've tried all 20, and here are the 100 different permutations, and and here's my results. And that type of thing, I think it will work very well. Type of scaling to once you solve one problem to to make the AI attack a 100 adjacent problems. The things that humans do still so so where the AI really struggles right now is knowing when it's made a wrong turn.
它可以说，哦，我要解决这个问题。我要把这个问题分解成这两种情况。我要尝试这种技术。有时候如果你幸运，而且问题简单，这就是正确的技术，你就能解决问题。有时候它会遇到问题，会提出一个完全荒谬的方法。
And it can say, oh, I'm gonna solve this problem. I'm gonna split up this problem into into these two cases. I'm gonna try this technique. And sometimes if you're lucky and it's a simple problem, it's the right technique and you solve the problem. Sometimes it will get it will have a problem with it it would propose an approach which is just complete nonsense.
但是，它看起来像个证明。这就是LLM生成的数学一个令人讨厌的地方。是的，我们也有过人类生成的非常低质量的数学，比如那些没有经过正规训练的人提交的内容。但如果一个人工证明很糟糕，你很快就能看出来。
And but, like, it looks like a proof. So this is one annoying thing about LLM generated mathematics. So Yeah. We we've we've had human generated mathematics that's very low quality, like, you know, submissions where we don't have the formal training and so forth. But if a human proof is bad, you can tell it's bad pretty quickly.
它会犯非常基本的错误，但AI生成的证明表面上看起来可能完美无缺。部分原因是强化学习实际上训练它们这样做，生成看起来正确的内容，这在许多应用中已经足够好了。所以错误通常非常微妙，当你发现它们时，它们又显得极其愚蠢。就像，你知道，没有人类会真正犯那种错误。
It makes really basic mistakes, but the AI generally proves they can look superficially flawless. And it's partly because that's what the reinforcement learning has actually trained them to do, to to make things to to produce text that looks like what is correct, which for many applications is good enough. So error is often really subtle, and then when you spot them, they're they're really stupid. Like, you know, like, no human would have actually made that mistake.
是的。在编程背景下这实际上非常令人沮丧，因为我经常编程。而且，当人类写出低质量代码时，有一种叫做代码异味的东西。对吧？你能看出来。
Yeah. It's actually really frustrating in the program context because I I program a lot. And, yeah, when a human makes when a low quality code, there's something called code smell. Right? You you could tell.
你能看出来。立刻，就像
You could tell. Immediately, like
哦，是的。
Oh, yeah.
有迹象。但是AI生成的代码
There's signs. But with with AI generated code
无味的。
The odorless.
然后你是对的。是的。最终，你会发现一个明显的愚蠢之处，看起来却像是好代码。是的。
And then you're right. Yeah. Eventually, you find an obvious dumb thing that just looks Yeah. Like good code. Yeah.
所以这也很棘手，而且出于某种原因令人沮丧
So It's very tricky too, and frustrating for some reason to
是的。所以必须努力。是的。所以嗅觉很重要。好吧。
Yeah. So have to work. Yeah. So the sense of smell. Okay.
就是这样。这是人类拥有的一个特质，而且存在一种隐喻性的、数学的嗅觉，目前还不清楚如何让AI复制这一点。最终，我的意思是，AlphaZero等系统在围棋和国际象棋等方面取得进展的方式，在某种意义上，它们已经发展出了对围棋和国际象棋局面的嗅觉。嗯，知道这个局面对白方有利，但对黑方不利。
There you go. This this is this is one thing that humans have, and there's there's a metaphocal, mathematical smell that this with it's not clear how to get the AIs to duplicate that. Eventually I mean, so the way alpha zero and so forth make progress on Go and and chess and so forth is is in some sense, they have developed a sense of smell for Go and chess positions. Mhmm. Know, that that this position is good for white, but it's good for black.
它们无法阐明原因，但仅仅拥有这种嗅觉就让它们能够制定策略。所以如果AI获得了评估某些证明策略可行性的能力，那么你就可以说，我要尝试将这个问题分解成两个较小的子任务，然后你可以说，哦，这看起来不错。这两个任务看起来比你的主要任务更简单，而且它们仍然有很大的可能是正确的。所以值得尝试。或者现在你把问题变得更糟了，因为两个子问题实际上都比原始问题更难，这通常是你尝试随机方法时会发生的情况。
They can't enunciate why, but just having that that sense of smell lets them strategize. So if AI's gained that ability to sort of assess the viability of certain proof strategies, so so you can say, I'm gonna try to to break up this problem into two small subtasks, and then you can say, oh, this looks good. The two tasks look like they're simpler tasks than than your main task, and they still got a good chance of being true. So this is good to try. Or now you've you've you made the problem worse because each of the two subproblems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try.
通常，实际上很容易将一个问题转化为更困难的问题。很少能将问题转化为更简单的问题。是的。所以如果它们能够获得这种嗅觉，那么也许就能开始与人类水平的数学家竞争了。
Normally, actually it's very easy to transform a problem into an even harder problem. Very rarely do you problem transport a simpler problem. Yeah. So if they can pick up a sense of smell, then they could maybe start competing with human level mathematicians.
所以这是个难题，但不是竞争，而是协作。是的。如果可以的话。假设性的。如果我给你一个预言机，它能做到你工作的某些方面，你可以直接与它合作
So this is a hard question, but not competing, but collaborating. Yeah. If okay. Hypothetical. If I gave you an oracle that was able to do some aspect of what you do, you could just collaborate with it
是的。
Yeah.
是的。是的。预言机，你希望这个预言机能做什么？你希望它，也许能作为一个验证器，嗯。检查代码异味之类的，你的是的。
Yeah. Yeah. Oracle what would you like that oracle to be able to do? Would you like it to, maybe be a verifier, like, Mhmm. Do the code smell like, your yes.
是的。Tawel教授，这是正确的这是一个好的这是一个有前途、富有成果的方向。
Yeah. Professor Tawel, this is the correct this is a good this is a promising, fruitful direction.
是的。是的。是的。
Yeah. Yeah. Yeah.
或者或者你希望它，生成可能的证明，然后你判断哪个是正确的？或者你希望它可能生成不同的表示方式，完全不同的看待这个问题的方法？是的。
Or 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, different totally different ways of seeing this problem? Yeah.
我认为以上所有。很大程度上是我们不知道如何使用这些工具，因为这是一种范式，它不是的。过去我们没有足够能力理解复杂指令的系统。嗯。它们可以大规模工作，但也不可靠。就像，它有趣地在细微之处有点不可靠，同时又能提供足够好的输出。
I think all of the above. A a lot of it is we don't know how to use these tools because it is a paradigm that it's not yeah. We have not had in the past systems that are competent enough to understand complex instructions Mhmm. That can work at massive scale, but are also unreliable. Like, it's it's an interesting a bit unreliable in subtle ways whilst we whilst providing sufficiently good output.
这是一个有趣的组合。你知道，我的意思是，你有一些研究生和你一起工作，他们有点像这样，但规模不大。你知道吗？而且我们之前有一些可以在大规模上工作的软件工具，但范围非常狭窄。所以我们必须弄清楚如何利用，我是说，就像蒂姆·卡尔提到的，亚基斯在二月份就预见到了，他当时就在设想数学在未来二十五年会是什么样子。
It's a interesting combination. You know, I mean, you have you have, like, graduate students that you work with who've, like, kinda like this, but not at scale. You know? And and and we had previous software tools that can work at scale, but but very narrow. So we have to figure out how to how to use I mean, so Tim Carr, like, you mentioned Yakis foresaw, like, in in February, he was envisioning what mathematics would look like in in actually two and a half decades.
这很有趣。是的。他在他的文章中写了一个假设性的对话，是关于未来的数学助手和他自己之间的对话，你知道，试图解决一个问题，他们需要进行对话。有时候人类会提出一个想法，AI会评估它。有时候AI会提出一个想法，有时候需要竞争，AI就会直接说，好吧。
That's funny. Yeah. He he wrote in his in his article, like, a a a hypothetical conversation between a mathematical assistant of the future and himself, you know, trying to solve a problem, and they would have to have a conversation. They sometimes the human would propose an idea, and the AI would would evaluate it. And sometimes the AI would propose an idea, and and sometimes a competition was required, and AI would just go and say, okay.
我已经检查了这里需要的100个案例。或者你先为所有n设置这个。我已经检查到n=100，目前看起来不错。等等，n=46时有个问题。
I've I've checked the the 100 cases needed here. Or the first you you set this for all n. I've checked n up to 100, and it looks good so far. Or hang on. There's a problem at n equals 46.
所以这只是一个自由形式的对话，你事先不知道事情会如何发展，但基于双方都认为好的想法。计算也是双方的事。我和AI有过这样的对话，我说，好吧，我们要合作解决这个数学问题，这是一个我已经知道答案的问题。所以我尝试引导它。
And so just a free form conversation where you don't know in advance where things are gonna go, but just based on on I think ideas are good for both sides. Calculations are for both sides. I've had conversations with AI where I say, okay. Let's we're gonna collaborate to solve this math problem, and it's a problem that I already know a solution to. So I I try to prompt it.
好吧，问题是这样的。我建议使用这个工具，但它会使用一个完全不同的工具，最终走向错误的方向，然后说，不，不，不。
Okay. So here's the problem. I I suggest using this tool, and it it'll this lovely arc in using a completely different tool, which eventually goes, you know, into the weeds and say, no. No. No.
试试用这个。好吧。它可能会开始使用这个，然后又回到我之前想要的那个工具。就像你必须不断把它拉回你想要的路径上。我最终可以强迫它给出我想要的证明，但这就像赶猫一样困难。
Try using this. Okay. And it might start using this, and then it'll go back to the tool that I wanted to to to to before. And, like, you have to keep railroading it onto the path you want. And I I could eventually force it to give the proof I wanted, but it was like herding cats.
就像我需要付出的个人努力不仅仅是引导它，还要检查它的输出，因为很多看起来会成功的东西实际上有问题。我知道在917上有个问题，基本上是在和它争论。这比没有辅助的情况下自己做还要累人。所以，这就是当前的技术水平。
Like and the amount of personal effort I had to take to not just sort of prompt it, but also check its output because it like, a lot of what it looked like is gonna work. I know there's a problem on nine seventeen, and basically arguing with it. Like, it was more exhausting than doing it unassisted. So, like, it but that's the current state of the art.
我在想是否会出现一个相变，到那时就不再感觉像在放牧一群猫一样困难了，也许它会以多快的速度到来让我们感到惊讶。
I wonder if there's there's a phase shift that happens to where it's no longer feels like herding cats, and maybe it'll surprise us how quickly that comes.
你知道吗？我相信会的。在形式化方面，我之前提到过，手工形式化一个证明需要花费10倍的时间。有了这些现代AI工具，以及更好的工具支持。Lean开发者们做得很好，不断增加更多功能并使其更加用户友好。
You know? I I believe so. So in formalization, I I mentioned before that it takes 10 times longer to formalize a proof of antibiotic by hand. With these modern AI tools, it's and also just better tooling. The lean developers are doing a great job adding more and more features and making it user friendly.
它会从九降到八再到七。好吧，没什么大不了的。但总有一天，它会下降一个数量级，那就是相变。因为突然间，当你写论文时，先写在Lean里或者通过与AI对话（它能实时为你生成Lean代码）就变得合理了。
It's going from nine to eight to seven. Okay. No big deal. But one day, it will drop a little one, and that's the phase shift. Because suddenly, it makes sense when you write a paper to to write it in Lean first or through a conversation with AI who's generating Lean on the fly with you.
而且期刊接受起来也变得自然。也许他们会提供加急审稿，你知道，如果论文已经在Lean中形式化了，他们只会请审稿人评论结果的重要性以及它与文献的联系，而不必太担心正确性，因为那已经被认证了。数学论文变得越来越长。实际上，为那些非常长的论文找到好的审稿越来越难，除非它们真的非常重要。这确实是个问题，而形式化正好在此时出现来解决这个问题。
And it becomes natural for journals to accept. You know, maybe they'll offer expedite refereeing, you know, that if if a paper has already been formalized in lean, they'll just ask the referee to comment on on the significance of the results and how it connects to literature and not worry so much about the correctness because it that's been certified. Papers are getting longer and longer in mathematics. It actually is harder and harder to get good refereeing for the the really long ones, unless they're really important. It is actually an an issue, which and the formalization is coming in just the right time for this to be.
而且由于工具和其他所有因素，猜测变得越来越容易，那么你会看到更多，比如数学标签的增长，对吧。可能是指数级的。是的，这是一个良性循环。好的。
And the easier and easier to guess because of the tooling and all the other factors, then you're gonna see much more, like, math label grow Right. Potentially exponentially. Yeah. It's a it's a it's a virtuous cycle. Okay.
我的意思是，过去发生的这类相变之一是LaTeX的采用。LaTeX是一种排版语言，现在所有数学家都在使用。过去，人们使用各种文字处理器、打字机等等。但在某个时刻，LaTeX变得比所有其他竞争对手都更容易使用，然后人们会在几年内就切换过来。就像，那是一个戏剧性的相变。
I mean, one phase shift of this type that happened in the past was the Adoption's LaTeX. So so LaTeX is is type setting language that all mathematicians use now. So in the past, people use all kinds of word processors and typewriters and and whatever. But at some point, LaTeX became easier to use than all other competitors, and, like, people would switch, you know, within a few years. Like, it was just a dramatic phase shift.
这是一个大胆的问题，但是哪一年？我们距离一个AI系统成为赢得菲尔兹奖的证明的合作者还有多远？
It's a wild out there question, but what what year? How far away are we from a AI system being a collaborator on a proof that wins the Fields Medal?
怎么做到的？
How?
就是那个水平。
So that level.
好吧。这取决于合作的程度。
Okay. Well, it depends on level of collaboration.
意思是，不。就像，它值得获得菲尔兹奖。差不多一半一半吧。
Mean No. Like, it deserves to be to get the Fields Medal. Like so half and half.
其实，我可以想象如果是一篇获奖论文，其中使用了某些AI系统来协助写作，你知道，就像，以前完全靠自己写已经很困难了，我现在就用AI。它加快了我自己的写作速度。比如，你有一个定理，还有一个证明，证明分为三种情况。我写下第一种情况的证明后，自动补全功能就会建议第二种情况的证明可以怎么做。而且，它完全正确。
Already, like, I I can imagine if it was for a medal winning paper having some AI systems in writing it, you know, just, you know, like, the old complete alone is already I I I use it. Like, it speeds up my my own writing. Like, you know, you you you can have a theorem, and you have a proof, and the proof has three cases. And I I I write down the proof of first case, and the autocomplete just suggests that now here's how the proof of second case could work. And, like, it was exactly correct.
那太棒了。为我节省了大概五到十分钟的打字时间。
That was great. Saved me, like, five, ten minutes of of of typing.
但在那种情况下，AI系统不会获得菲尔兹奖。
But in that case, the AI system doesn't get the Fields Medal.
不。
No.
我们说的是二十年、五十年还是一百年？你怎么看？
Are we talking twenty years, fifty years, a hundred years? What do you think?
好的。我在出版物中做过一个预测，到2026年，也就是明年，将会出现与AI的数学合作。不是菲尔兹奖级别的，而是真正的研究级数学。
Okay. So I I gave a prediction in print, so by 2026, which is now next year, there will be math collaborations, you know, with AI. So not field's medal winning, but but, like, actual research level math
比如，那些...部分由AI生成的想法？可能不是想法本身，
Like, ideas that Yeah. In part generated by AI? Maybe not the ideas,
但至少是一些计算、验证工作。是的，我是说，
but at least some of the computations, the verifications. Yeah. I mean,
这...这已经发生了吗？
there there Has that already happened?
这已经发生了吗？是的。有些问题是通过与AI复杂对话来解决的，AI提出建议，然后人类去尝试，但合同不成立。不过它可能会提出不同的想法。很难完全理清其中的界限。
Has that already happened? Yeah. There there are there are problems that were solved by a complicated process converse conversing with with AI to propose things, and then the human goes and tries it, and and the contract doesn't work. But the the it might pose a different idea. It it's it's hard to disentangle exactly.
确实存在一些数学成果，只有通过数学人类和AI的合作才能完成。但很难完全分清各自的功劳。我的意思是，这些工具并不能复制进行数学研究所需的全部技能，但它们能够复制其中相当一部分，比如40%左右。所以它们能够填补空白。编程就是一个很好的例子。
There are certainly math results which could only have been accomplished because there was a math math human and an AI involved. But it's hard to sort of disentangle credit. I mean, these tools, they they do not replicate all the skills needed to do mathematics, but they can replicate sort of some nontrivial percentage of them, you know, 40%. So they can fill in gaps, you know. So coding is is is a is a good example.
你知道吗？对我来说用Python编程很烦人。我不是专业的程序员。但有了AI，进行编程的摩擦成本大大降低了。所以它为我填补了这个空白。
You know? So I I it's annoying for me to to code in Python. I'm not I'm not a native, you know, professional programmer. But the with AI, the the the the friction cost of of doing it is is is much reduced. So it it fills in that gap for me.
AI在文献综述方面已经相当不错了。虽然还存在幻觉问题，会编造不存在的参考文献。但我认为这是个可以解决的问题。如果采用正确的训练方式等，并且通过互联网进行验证，几年后应该能达到这样的程度：当你需要一个引理时，可以问'有人证明过这个引理吗？'
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 civil war problem. If you train in the right way and so forth, you can you can and and verify, you know, using the Internet. You you know, you should, in a few years, get the point where you you have a a lemma that you need and say, has anyone proven this lemma before?
它基本上会进行一种高级的网络搜索AI辅助。是的。它会给出六篇论文，其中可能有类似的情况发生。我的意思是，你现在问我，它可能会给出六篇论文，其中也许只有一篇是合法且相关的，一篇存在但不相关，另外四篇都是幻觉产生的。
And it will do basically a fancy web search AI assistance. Yeah. Yeah. There are these six papers where something similar has happened. And I mean, you can ask me right now, and it'll give you six papers of which maybe one is legitimate and relevant, one exists but is not relevant, and four are hallucinated.
目前它确实有一定的成功率，但垃圾信息太多，信噪比太差，所以它最有帮助的时候是你已经对某个领域有所了解，只需要被提醒想起那些已经潜藏在记忆中的论文。
It has a nonzero success rate right now, but it's there's so much garbage, so much the signal to noise ratio is so poor that it's it's it's most helpful when you already somewhat know the relationship, and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.
而不是帮助你发现你完全不知道但却是正确引用的新文献。
Versus helping you discover new you were not even aware of, but is the correct citation.
是的。它有时确实能做到这一点，但当它做到时，这个正确的结果往往被埋在一堆糟糕的选择中。
Yeah. That's yeah. That it can sometimes do, but but when it does, it's it's buried in in a list of options for which the other bad.
是的。我是说，能够自动生成一个正确无误的相关工作章节，这确实是一件美妙的事情，可能会带来另一个阶段性的转变，因为它能正确地归功。是的，它确实打破了信息孤岛的壁垒。
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 credit correctly. Yeah. It does it breaks you out of the silos of.
是的。是的。是的。不。思考。
Yeah. Yeah. Yeah. No. Thought.
你知道吗？是的。不。但目前有一个巨大的障碍需要克服。我的意思是，这就像自动驾驶汽车一样。
You know? Yeah. No. But it it there's a big hump to overcome right now. I mean, it's it's it's like self driving cars.
对。你知道，安全边际必须非常高，是的，这样才能可行。所以，许多人工智能应用都存在最后一英里问题，你知道，有些工具在20%、80%的情况下能工作，但仍然不够好，甚至在某些方面比好的还要糟糕。
Right. You know, the the safety margin has to be really high Yeah. For it to be to be feasible. So So there's a last mile problem with a lot a lot of AI applications that, you know, they can do there are tools that work 20%, 80% of the time, but it's still not good enough, and in fact, worse than good in some ways.
我是说，问菲尔兹奖问题的另一种方式是，你认为在哪一年你会醒来，感到真正的惊讶？你读到头条新闻，AI做了某件事，比如真正的突破。不一定是菲尔兹奖或甚至假设，它可能就像AlphaZero那样的时刻。
I mean, another way of asking the Fields Medal question is, what year do you think you'll wake up and be, like, real surprised? You read the headline, the news of something happened that AI did, like, you know, real breakthrough, something. It doesn't you know, like, Fields Medal or even hypothesis, It could be, like, really just this alpha zero moment would go that kind
那样的事情。对。是的。这个十年，我可以预见它，比如在两个人们认为无关的事物之间提出一个猜想？
of thing. Right. Yeah. This this decade, I can I can see it, like, making a conjecture between two unrelated two two things that people thought was unrelated?
哦，有趣。生成一个猜想，那是一个美妙的猜想。
Oh, interesting. Generating a conjecture, that's a beautiful conjecture.
是的。而且实际上，真正的信任已经被证明是正确的。
Yeah. And and actually has a real trust has been correct
而且而且而且有意义。因为那实际上有点可行，我想。但是数据在哪里，它确实是的。是的。不。
and and and meaningful. Because that's actually kind of doable, I suppose. But the where the data is, it's yeah. Yeah. No.
那将是真正令人惊叹的。
That would be truly amazing.
当前的模型确实很吃力。我的意思是，这个的一个版本是，物理学家们有一个梦想，让AI去发现新的物理定律。你知道，梦想就是你只需输入所有这些数据。然后这里有一个我们以前没见过的新的模式。
The current models struggle a lot. I mean, so a version of this is I mean, the the physicists have a dream of getting the AI to discover new new laws of physics. You know, the the dream is you just feed it all this data. Okay? And and this is a here here is a new pattern that we didn't see before.
但实际上，即使是当前最先进的技术，也很难从数据中发现旧的物理定律。我的意思是，或者即使它做到了，也存在一个很大的污染担忧，它之所以能做到，只是因为，比如在训练数据中的某个地方，已经有人知道了，你知道，玻意耳定律或者你要重建的任何东西。部分原因是我们没有适合这种任务的正确类型的训练数据。是的。所以对于物理定律，我们没有，比如，一百万个不同的宇宙，每个宇宙有一百万种不同的自然法则。
But it actually even struggle the current state of the art even struggles to discover old laws of physics from the data. I mean, or if it does, there's a big concern of contamination that that it did only because, like, somewhere in this training data, it already someone knew, you know, Boyle's law or whatever you're to to to reconstruct. Part of it is that we we don't have the right type of training data for this. Yeah. So for laws of physics, like, we we don't have, like, a million different universes with a million different balls of nature.
而且，我们在数学中缺少的很多东西实际上是负面空间——我们有已经发表的东西，人们能够证明的定理，以及最终被验证的猜想，或者产生了反例。但我们没有关于那些被提出、看起来值得尝试，但人们很快意识到是错误的猜想的数据。然后他们说，哦，但我们应该改变我们的主张，以这种方式修改它，使其更合理。有一个试错的过程，这是人类数学发现中真正不可或缺的一部分，我们没有记录它，因为它令人尴尬。我们会犯错，而我们只喜欢发表我们的成功。
And, like, 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 would be counterexamples produced. But we don't have data on 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 they said, oh, but we we should actually change our claim to modify it in in this way to actually make it more plausible. There's 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 it's embarrassing. We make mistakes, and and we only like to publish our our wins.
而AI无法访问这些数据进行训练。我有时开玩笑说，基本上，AI必须去读研究生，实际上，你知道，去上研究生课程，做作业，去办公室时间，犯错误，得到如何纠正错误的建议，并从中学习。
And the AI has no access to this data to train on. I sometimes joke that basically, we had to get AI has to go through 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.
请允许我问您关于格里戈里·佩雷尔曼的问题。嗯。您提到在工作中会尽量保持谨慎，不让一个问题完全占据您的生活。只是您真的会爱上这个问题，不解决它就无法安心。但您也急忙补充说，有时候这种方法实际上可以非常成功。
Let me ask you, if I may, about Grigori Perlman. Mhmm. You mentioned that you try to be careful in your work and not let a problem completely consume you. Just you've really fallen in love with the problem and really cannot rest until you solve it. But you also hasted to add that sometimes this approach actually can be very successful.
嗯。您举的例子是格里戈里·佩雷尔曼，他证明了庞加莱猜想，并且是通过独自工作七年，几乎不与外界接触的方式完成的。您能解释一下这个已被解决的千禧年大奖难题——庞加莱猜想，并谈谈佩雷尔曼所经历的旅程吗？
Mhmm. An example you gave is Gurgorye Perlman, who proved the Poincare 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, Poincare conjecture, and maybe speak to the journey that Kugura Perlman's been on.
好的。这是一个关于弯曲空间的问题。这是个很好的例子。我当时想的是一个二维曲面。在它周围可能是一个带洞的环面，或者可以有多个洞。
Alright. So it's it's a question about curved spaces. That's a good example. So I was thinking it was a two d surface. In just around the it could maybe be torus with a hole in it or it can have many holes.
而且即使假设曲面是有界且光滑的等等，从先验角度来看，曲面也可能具有许多不同的拓扑结构。我们已经找到了分类曲面的方法。作为初步近似，一切都是由所谓的亏格决定的，也就是它有多少个洞。所以球面的亏格为零，甜甜圈的亏格为一，依此类推。区分这些曲面的一个方法是，球面具有所谓的单连通性。
And there there are many different topologies a priori that that a surface could have, even if you assume that it's it's bounded and and smooth and so forth. So we've we have figured out how to classify surfaces. As a first approximation, everything's determined by some called the genus, how many holes it has. So the sphere has genus zero, a doughnut has genus one and so forth. And one way you can tell these surfaces apart, probably the sphere has which is called simply connected.
如果你在球面上取任意闭合环，比如一个大闭合绳圈，你可以在保持在表面的同时将其收缩到一个点。球面具有这个性质，但环面没有。如果你在环面上，取一根绕行环面外径的绳子，它无法穿过那个洞，无法收缩到一个点。所以事实证明，球面是唯一具有这种可收缩性质的曲面，我的意思是在球面的连续变形范围内。
If you take any closed loop on the sphere, like a big closed loop of 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. And if you're on a torus and you take a rope that goes around, say, the the outer diameter of torus, there's no way it it can't get through the hole. There's no way to to contract it to a point. So it turns out that the the the sphere is the only surface with this property of contractability, I mean, up to, like, continuous deformations of the sphere.
所以这些就是我们称为拓扑等价于球面的东西。庞加莱猜想在更高维度提出了同样的问题。这变得难以可视化，因为曲面可以想象为嵌入三维空间中。但作为一个弯曲的自由空间，我们对四维空间没有很好的直觉来居住。而且还有一些三维空间甚至无法嵌入四维空间。
So so things that are what we call topologically equivalent to the sphere. So point QA asked the same question in higher dimensions. So this it becomes hard to visualize because surface you can think of as embedded in three dimensions. But as a curved free space, we don't have good intuition of four d space to to to live in. And then there there are also three d spaces that can't even fit into four dimensions.
需要五维、六维或更高维度。但无论如何，数学上你仍然可以提出这个问题：如果你有一个有界的三维空间，它也具有这种单连通性质——每个环都可以收缩，你能把它变成三维版本的球面吗？这就是庞加莱猜想。奇怪的是，在更高维度（四维和五维）中，它实际上更容易解决，所以首先在高维中得到了解决。
You need five or six or or higher. But anyway, mathematically, you can still pose 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 sphere? And so this is the Poincare conjecture. Weirdly, in higher dimensions, four and five, it was actually easier. So it was solved first in high dimensions.
不知为何，进行形变的空间似乎更大了。更容易将物体移动到你想要的球体形状。但三维情况确实非常困难。因此人们尝试了许多方法。有一种组合方法，将表面切割成小三角形或四面体，然后试图通过面与面之间的相互作用来进行论证。
There's somehow more room to do the deformation. It is easier to to to move things around to to to your sphere. But three was really hard. So people tried many approaches. There's sort of combinatory approaches where you chop up the the the surface into little triangles or or tetrahedra, and you you you just try to argue based on how the faces interact each other.
还有代数方法。存在各种可以称为基本群的代数对象，你可以附加这些同调和上同调以及所有这些非常花哨的工具。它们也不太奏效。但理查德·汉密尔顿提出了一种偏微分方程方法。问题是，你有一个本质上其实是球体的对象，但它以一种非常奇怪的方式呈现给你。
There were algebraic approaches. There's there's various algebraic objects that can called the fundamental group that you can attach to these homology and cohomology and and and and all these very fancy tools. They also didn't quite work. But Richard Hamilton's proposed a partial differential equations approach. So you take you take so the problem is that you're so you have this object which is so secretly the sphere, but it's given to you in a in a really in a in a weird way.
就像我想象一个被揉皱扭曲的球，但看不出它原本是个球。但如果你有一个变形的球面，比如气球表面，你可以尝试给它充气。你吹气进去，随着空气填充，皱纹会逐渐平滑，最终变成一个漂亮的圆球。
So it's like I think of a ball that's been kind of crumpled up and twisted, and it's not obvious that it's a ball. But, like, if you if you have some sort of surface which is which is a deformed sphere, you could you could for example, think of that as a surface of a balloon. You could try to inflate it. You you blow it up. And naturally, as you fill the air, the the wrinkles will sort of smooth out, and it will turn into a nice round sphere.
是的。当然，除非它是个环面或其他形状，那样就会在某个点卡住。比如如果是环面，中间会有一个点。当内环收缩到零时，就会出现奇点，无法再继续吹气，无法继续流动。
Yeah. Unless, of course, it was a torus or something, which case it would get stuck at some point. Like, if you instead of torus, there it would there'll there'll be a a point in the middle. When the inner ring shrinks to zero, you get a you get a singularity, and you you can't blow up any further. You can't flow any further.
于是他创建了这个流，现在称为里奇流，这是一种处理任意表面或空间并使其越来越圆滑，使其看起来像球体的方法。他想证明这个过程要么会得到一个球体，要么会产生奇点。这很像偏微分方程要么具有全局正则性，要么在有限时间内爆破。基本上，这几乎是完全相同的概念，都是相互关联的。
So he created this float, which is now called Ricci flow, which is a way of taking an arbitrary surface or or space and smoothing it out to make it rounder and rounder, to make it look like a sphere. And he wanted to show that that either this process would give you a sphere or it would create a singularity. I can very much like how PDEs either have they have global regularity or finite hand blow up. Basically, it's almost exactly exactly the same thing. It's all connected.
他证明了对于二维曲面，如果开始时是单连通的，就不会形成任何奇点。你永远不会遇到麻烦，可以持续流动，最终得到一个球体。嗯。因此他为二维结果提供了一个新的证明。
And so and and he showed that for two dimensions, two dimensional surfaces, if you start a symphony connected, no singularity is ever formed. You you never ran into trouble, and you could flow, and and it will give you a spear. Mhmm. And it so he he got a new proof of the two dimensional result.
顺便说一下，这个关于流动应用及其背景的解释非常精彩。这里的数学难度如何，比如二维情况？是不是很困难？
But by the way, that's a beautiful explanation where we should flow in its application and its context. How difficult is the mathematics here, like, for the two d case? Is it Yeah.
这些方程相当复杂，与爱因斯坦方程水平相当，虽然稍微简单一些，但确实如此。但它们被认为是难以求解的非线性方程。在二维中有很多特殊技巧有所帮助。但在三维中，问题在于这个方程实际上是超临界的。这与纳维-斯托克斯方程面临的是同样的问题。
These are quite sophisticated equations on par with the Einstein equations of slightly simpler, but yeah. But but they were considered hard nonlinear equations to solve. And there's lots of special tricks in two d that that that helped. But in three d, the problem was that this equation was actually supercritical. It's the same problem as as Navier Stokes.
随着尺度放大，曲率可能会集中在越来越小、越来越精细的区域，看起来越来越非线性，情况变得越来越糟糕。可能会出现各种奇点。有些奇点，比如所谓的颈缩现象，表面会像杠铃一样，在某一点收缩。有些奇点足够简单，你可以知道下一步该怎么做。只需做一个切割，就能将一个表面分成两个，然后分别处理。
As you blow up, maybe the curvature could get concentrated in finer and smaller smaller regions, and it it looked more and more nonlinear, and things just look worse and worse. And there could be all kinds of singularities that showed up. Some singularities, like, if you there's these things called neck pinches where where the the surface sort of creates behaves like a like a like a barbell, and it pinches at a point. Some some singularities are simple enough that you can sort of see what to do next. You just make a snip, and then you can turn one surface into two and you build them separately.
但问题是可能会出现一些非常棘手的、打结的奇点，你完全不知道如何解决，无法进行任何手术操作。所以需要对所有奇点进行分类。也就是要弄清楚所有可能出错的方式。所以佩雷尔曼做的是，首先他将这个超临界问题转化为了一个临界问题。我之前提到过能量的发明，即哈密顿量，它真正澄清了牛顿力学。
But those those are the the prospect that there's some really nasty, like, knotted singularities showed up that you you couldn't see how to resolve in any way, that you couldn't do any surgery to. So you need to classify all the singularities. Like, what are all the possible ways that things can go wrong? So what Kraumann did was first of all, he he made the problem he turned the problem with supercritical problem to a critical problem. I said before about how the invention of the of of energy, the Hamiltonian, that really clarified Newtonian mechanics.
于是他引入了一些现在被称为佩雷尔曼约化体积和佩雷尔曼熵的概念。他引入了新的量，有点像能量，在每个尺度上看起来都一样，将问题转化为临界问题，使得非线性突然看起来比以前不那么可怕了。然后他仍然需要分析这个临界问题的奇点。这本身就是一个类似的问题，实际上我看到过有人研究这个。就难度级别而言，他成功分类了这个问题的所有奇点，并展示了如何对每个奇点进行手术，通过这种方式，最终解决了庞加莱猜想。
So he introduced something which is now called Perlmens reduced volume and Perlmens entropy. He introduced new quantities kinda like energy that look the same at every single scale and 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. And that itself was a problem similar to this way I've seen it worked on, actually. So on the 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 to resolve the point grade conjecture.
所以这需要很多非常雄心勃勃的步骤，比如今天的大型语言模型最多只能把这个想法作为数百种尝试方案之一提出来。但其他99种可能都是死胡同，你只有在数月工作后才能发现。他一定是有某种直觉，认为这是正确的方向，因为你知道，从A到B需要花费数年时间。
So quite like, a lot of really ambitious steps, and like 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'd only find out after months of of work. He must have had some sense that this was the right track to pursue because, you know, it takes years to get them from a to b.
所以就像你说的，实际上，即使在严格的数学意义上，但更广泛地
So you've done, like you said, actually, see even strictly mathematically, but more broadly in
来说
terms
在这个过程中，你也做过类似困难的事情。从他独自经历的这个过程中，你能推断出什么？因为他是一个人做的。像你提到的困难那样，当AI不知道自己在失败时，这样的过程中会有哪些低谷？你会发生什么？
of the process, you've done similarly difficult things. What 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 hardship, like, AI doesn't know when it's failing. What what happens to you?
当你坐在办公室里，意识到自己过去几天甚至几周做的事情都是失败的。
You're sitting in your office when you realize the thing you did for the last few days, maybe weeks Yeah. Is a failure.
对我来说，我会转向不同的问题。就像我说的，我是只狐狸，不是刺猬。
Well, for me, I switched to different problem. So as I said, I'm I'm I'm I'm a fox. I'm not a hedgehog.
但你通常可以采取的休息方式就是暂时放下，看看不同的问题。是的。
But you generally, that is a break that you can take is is to step away and look at a different problem. Yeah.
你也可以修改问题。我的意思是，如果某个特定问题阻碍了你，某个糟糕的情况不断出现，而你的工具对此无效，你可以直接假设这个糟糕情况不会发生。所以你进行一些魔法思维，但从战略上来说，这是为了看看论证的其余部分是否成立。如果你的方法存在多个问题，那么也许你就该放弃了。
You can modify the problem too. I mean, yeah, you can ask them if if there's a specific thing that's blocking you that that this some bad case keeps showing up that that that for which your tool doesn't work, you can just assume by fear this this bad case doesn't occur. So you you do some magical thinking for the you know, but but but strategically, okay, for the point to see if the rest of the argument goes through. If there's multiple problems with with with your approach, then maybe you just give up. Okay?
但如果这是我们知道的唯一问题，其他一切都检查无误，那么仍然值得继续努力。是的，有时候你需要进行某种前瞻性侦察，假设'好吧，我们会解决的'，这有时是富有成效的。
But if this is the only problem that we know, then everything else checks out, then it's still worth fighting. So yeah. Yeah. You have to do some some sort of forward reconnaissance sometimes to to, you know And that is sometimes productive to assume, like, okay. We'll figure it out.
哦，是的。是的。最终。有时候，实际上犯错甚至也是富有成效的。
Oh, yeah. Yeah. Yeah. Eventually. Sometimes, actually, it's it's even productive to make mistakes.
所以，我的意思是，有一个项目我们实际上还获了奖。其实，我和其他四个人一起合作。我们研究了这个PDE问题。又是一个关于爆破正则性的问题。而且这个问题被认为非常困难。
So one of the I mean, there's a project which actually we won some prizes for. Actually, for I I have four other people. We worked on this PDE problem. Again, actually, blow off regularity type problem. And it it was considered very hard.
Jean Bourguin是另一位领域专家，他研究了这个问题的特殊情况，但没能解决一般情况。我们花了两个月研究这个问题，以为我们解决了。我们有一个巧妙的论证，如果一切成立，我们都很兴奋。我们计划庆祝，聚在一起喝香槟什么的。然后我们开始撰写论文，其中一位合著者（不是我）说，在这个引理中，我们需要估计这个展开式中出现的13个项。
Jean Bourguin was another field specialist who worked on a special case of this, but he could not solve the general case. And we worked on this problem for two months, and we found we thought we solved it. We we had this this cute argument that if anything fit, we were excited. We were planning celebration to all get together and have champagne or something. And we started writing it up, and one of one of us, not me, actually, but another co author said, oh, in this in this lemma here, we we have to estimate these 13 terms that that show up in this expansion.
我们估计了其中的12项，但在我们的笔记中，我找不到第13项的估计，你们能找到吗？有人能提供吗？我说，当然，我来看看这个。我会处理的。
And we estimate 12 of them, but in our notes, I can't find the the estimation of the thirteenth, can you? Can someone supply that? And I said, sure. I'll look at this. And I'll actually yeah.
我们根本没处理那一项。我们完全遗漏了这个项。而这个项的结果比另外12项加起来还要糟糕。事实上，我们无法估计这个项。我们又尝试了几个月，试了所有不同的排列组合，但总是有这个无法控制的项。
We didn't cover that. We completely omitted this term. And this term turned out to be worse than the other 12 terms put together. In fact, we could not estimate this term. And we tried for a few more months and and all different permutations, and there was always this one thing one term that we could not control.
所以，这非常令人沮丧。但因为我们已经投入了数月努力，我们坚持了下来，尝试了越来越绝望和疯狂的方法。两年后，我们找到了一个与初始策略截然不同的方法，这个方法没有产生这些有问题的项，最终解决了问题。所以我们花了两年解决了问题。但如果我们没有最初几乎解决问题的错误突破，我们可能在第二个月就放弃去研究更简单的问题了。
And so, like, this was very frustrating. But because we had already invested months and months of effort in this already, we stuck at this, which we tried increasingly desperate things and and crazy things. And after two years, we found that approach was somewhat different by quite a bit from our initial strategy, which did actually didn't generate these problematic terms and and and actually solve the problem. So we we solve a problem after two years. But if we hadn't had that initial full storm of nearly solving a problem, we would have given up by month two or something and and worked on an easier problem.
是的。如果我们知道要花两年时间，不确定我们是否会开始这个项目。有时候，错误的认知其实就像哥伦布航行到新奥尔良（注：实际应为误解地球大小导致发现美洲）。他对地球大小的测量是错误的。
Yeah. If we had known it would take two years, not sure we would have started the project. Yeah. Sometimes actually having the incorrect you know, it's it's like Columbus traveling in New York, New Orleans. He had an incorrect version of measurement of the size of the earth.
他以为会发现通往印度的新贸易路线。至少他的计划书是这么说的。我的意思是，也有可能他其实心里知道真相。
He thought he was going to find a new new trade route to India. Or at least that was how he sold it in his prospectus. I mean, it it could be that he actually secretly knew.
但就心理层面而言，你是否会有那种情绪上的或自我怀疑的感觉几乎将你淹没？你知道吗？因为这些东西，数学是如此令人全神贯注，以至于它可能会击垮你。当你在一道问题上投入了那么多自己，结果却是错的，你可能会开始以类似的方式崩溃，就像国际象棋曾击垮过一些人一样。
But Just on the psychological element, do you have, like, emotional or, like, self doubt that just overwhelms you most like that? You know? Because it this stuff, it feels like math is is so engrossing that, like, it can break you. When you, like, invest so much yourself in the problem and then it turns out wrong, you could start to similar way, chess has broken some people.
是的。我认为不同的数学家对他们所做的事情有着不同程度的情感投入。我的意思是，对有些人来说，这只是一份工作。你知道，你有一个问题，如果解决不了，你就会转向下一个。是的。
Yeah. I I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people, it's just a job. You know, you you have a problem, and if it doesn't work out, you you will you go on the next one. Yeah.
所以事实上你总是可以转向另一个问题，这减少了情感上的联系。我的意思是，确实存在一些情况，你知道吧？所以有些问题是我们所谓的执念或顽疾，有些人就是会死死抓住那一个问题，年复一年地只思考那一个问题。然后，你知道，也许职业生涯会受到影响等等。他们会说，哦，但一旦我解决了这个问题，这个巨大的成功将弥补我所有这些年失去的机会。
So the fact that you can always move on to another problem, it reduces the emotional connection. I mean, there are cases you know? So there are certain problems that are what we call backlash or diseases where where where just latch on to that one problem, and they spend years and years thinking about nothing but that one problem. And, you know, maybe the the career suffers and so forth. Say, oh, but how could this big win this will you know, once I once I finish this problem, I I will make up for all the years of of of lost opportunity.
然后，那……我的意思是，偶尔偶尔会成功，但我真的不建议那些没有足够毅力的人这样做。是的。所以我从未对任何一个问题超级投入。有帮助的一点是，我们不需要预先承诺一定能解决我们的问题。嗯，当我们写项目申请书时，我们某种程度上会说我们将研究这一组问题。
And then that's that's I mean, occasionally occasionally, it works, but I I really don't recommend it for people who have the the right fortitude. Yeah. So I 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. Well, when we do grant proposals, we sort of say we will we will study this set of problems.
但即使我们不承诺，肯定在五年内，我会给出所有这些问题的证明。你知道吗？你承诺的是取得一些进展或发现一些有趣的现象，也许你没有解决那个问题，但你发现了一些相关的问题，你能对其提出一些新的见解。而那是一个可行得多的任务。
But even though we don't promise, definitely by five years, I will supply a proof of all these things. You know? We 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 you you can say something new about. And that's that's a much more feasible task.
但我相信对你来说也存在这样的问题。你在数学史上一些最困难的问题上已经取得了如此大的进展。那么，是否存在一个问题一直困扰着你，潜伏在黑暗的角落里，你知道，比如孪生素数猜想、黎曼假设、哥德巴赫猜想。
But I'm sure for you there's problems like this. You have you have made so much progress towards the hardest problems in the history of mathematics. So is there is there a problem that just haunts you, sits there in the dark corners, you know, twin prime conjecture, Riemann hypothesis, Globach conjecture.
孪生素数，那听起来……嗯，再说一次，我的意思是，像黎曼假设这样的问题，它们目前还遥不可及。
Twin prime, that sounds well, again so, I mean, the problem is like a Riemann hypothesis, those are so far out of reach.
我 我 我 你这么认为吗？
I I I Do you think so?
是的。甚至没有可行的策略。就像，即使我激活了这本书里我知道的所有作弊手段，还是没办法让我做到。是的。我觉得这需要数学另一个领域的突破先发生，然后有人意识到那会是引入这个问题的一个有用方法。
Yeah. There there's no even viable strategy. Like, even if I activate all my all the cheats that I know of in this book, like, it it there's just still no way to get me to be. Yeah. Like, it's it's I think it needs a breakthrough in another area of mathematics to happen first and for someone to recognize that it that would be a useful thing to transport into this problem.
所以也许我们应该稍微退后一步，就谈谈质数。
So we we should maybe step back for a little bit and just talk about prime numbers.
好的。
Okay.
所以它们通常被称为
So they're often referred to
数学的原子。你能谈谈这些原子提供的结构吗？自然数有两个基本运算附加在它们身上，加法和乘法。所以如果你想生成自然数，你可以做两件事中的一件。你可以从1开始，一遍又一遍地加1，这样就生成了自然数。
as the atoms of mathematics. Can you just speak to the structure that these atoms provide? The natural numbers have two basic operations attached to them, addition and multiplication. So if you wanna generate the natural numbers, you can do one of two things. You can just start with one and add one to itself over and over again, and that generates you the natural numbers.
所以，从加法角度看，它们很容易生成。一、三、四、五。或者你可以取质数。如果你想通过乘法生成，你可以取所有质数二、三、五、七，然后把它们全部乘起来。它们一起就给出了所有自然数，可能除了1以外。
So, additively, they're very easy to generate. One, three, four, five. Or you can take the prime number. If you wanna generate multiplicatively, you can take all the prime numbers two, three, five, seven, and multiply them altogether. And together, they they they gives you all the the natural numbers except maybe for one.
所以从加法角度和更特定的角度来看，自然数有两种不同的思考方式。分开来看，它们并不那么糟糕。比如，任何只涉及加法的自然数问题都相对容易解决，任何只涉及乘法的问题也相对容易解决。但令人沮丧的是，当你将两者结合起来时，突然就变得极其丰富——我的意思是，我们知道数论中确实存在不可判定的命题。
So there are these two separate ways of thinking about the natural numbers from an additive point of view and a more particular point of view. And separately, they're not so bad. So, like, any question about that a natural was only was addition is relatively easy to solve. And any question that only was modification is relatively easy 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 are statements in number theory that are actually as undecidable.
存在某些多项式和一些变量。它们在自然数中是否有解？答案取决于一个不可判定的陈述，比如数学公理是否一致。但即使是最简单的组合问题，比如将素数（某种乘法性质）与加二移位（某种加法性质）结合起来，分开来看我们对两者都很了解，但如果你问：当你将素数加二时，你能得到另一个素数的频率有多高？
There are certain polynomials and some number of variables. Is there a solution in the natural numbers? And the answer depends on on an undecidable statement, like like, whether the axioms of of methods are consistent or not. But, yeah, but even the the simplest problems that combine something more plagiarative such as the primes with something additive such as shifting by two. Separately, we understand both of them well, but if you ask, when you shift the prime by two, do you can you get a how often can
你能得到另一个素数吗？我们一直难以将两者联系起来，这令人惊讶地困难。我们应该说明，孪生素数猜想正是假设存在无限多对相差为二的素数。是的。有趣的是，你在推动这个领域回答这类复杂问题方面非常成功，比如你提到的格林-陶定理。
you get another prime? We it's been amazingly hard to relate the two. And we should say that the twin prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by two. Yes. 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.
它证明了素数包含任意长度的算术级数。对吧。能证明这样的事情真是令人震惊。
It proves that prime numbers contain arithmetic progressions of any length. Right. Which is mind blowing that you could prove something like that.
对。是的。通过这类研究，我们意识到不同的模式具有不同水平的不可破坏性。孪生素数问题之所以困难，是因为如果你取世界上所有的素数，比如3、5、7、11等等，其中存在一些孪生素数。11和13就是一对孪生素数，以此类推。
Right. Yeah. So what we've realized because of the this this type of of research is that this different patterns have different levels of indestructibility. So so what makes the twin prime problem hard is that if you take all the primes in the world, you know, three, five, seven, eleven, so forth, there are some twins in there. Eleven and thirteen is a twin prime, pair of twin primes and so forth.
但如果你愿意，可以轻易地删改素数以消除这些孪生素数。比如，孪生素数会出现，而且有无限多个，但它们实际上相当稀疏。我的意思是，最初有不少，但一旦到了百万、万亿级别，它们变得越来越罕见。实际上，如果有人能访问素数数据库，只需在这里那里编辑掉几个素数，通过仅移除约0.01%的素数（精心选择以实现这一点），就能使孪生素数猜想不成立。
But you could easily, if you wanted to, redact to the primes to get rid of to get rid of the these twins. Like, twins, they'd show up, and there are infinitely many of them, but they're actually reasonably sparse. Not there's there's not I mean, initially, there's quite a few, but once you got to the millions, trillions, they become rarer and rarer. And you could actually just you know, like, if if if someone was given access to the database of primes, you just edit out a few a few primes here and there, They could make the trim package false by just removing, like, point 01% of the primes or something. Just well well chosen to to to do this.
因此，你可以呈现一个经过审查的素数数据库，它通过所有素数的统计检验，遵守素数的各种性质，但不再包含任何孪生素数。这对孪生素数猜想是一个真正的障碍。意味着任何在实际素数中寻找孪生素数的证明策略，在应用于这些稍加编辑的素数时必定失败。因此，这必须是素数某种非常微妙精细的特征，无法仅从聚合统计分析中获得。
And so you could present a censored database of the primes, which passes all of the statistical tests of the primes. You know, that it it it obey things like the primes other things about the primes, but doesn't contain any trim primes anymore. And this is a real obstacle for the trim primes conjecture. Means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes. And so it it it must be some very subtle delicate feature of the primes that you can't just get from, like like, aggregate statistical analysis.
好的。所以那个排除了。是的。
Okay. So that's out. Yeah.
另一方面，算术级数被证明要稳健得多。比如，你可以取质数，实际上可以排除99%的质数。你知道吗？你可以任意选取90%的质数。我们证明的另一件事是，你仍然会得到算术级数。
On the other hand, athletic progressions has turned out to be much more robust. Like, you can take the primes, and you can eliminate 99% of the primes, actually. You know? And you can take take any 90% that you want. And it turns out and another thing we proved is that you still get asthmatic progressions.
算术级数非常，你知道，它们就像蟑螂一样顽强。
Athmatic progressions are much you know, they're like cockroaches.
任意长度的。是的。是的。这太疯狂了。我的意思是，对于不了解算术级数的人来说，它是指数字之间相差某个固定数值的序列。
Of arbitrary length. Yes. Yes. That's crazy. I mean, so so this for for people who don't know arithmetic progressions is a sequence of numbers that differ by some fixed amount.
是的。但这又像是无限猴子定理的现象。对于任何固定长度的集合，你无法得到任意长度的级数。你只能得到相当短的级数。
Yeah. But it's again like it's it's infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary length progressions. You only get quite short progressions.
但你说孪生质数不是无限猴子现象。我的意思是，这是一个非常微妙的现象——它仍然属于无限猴子现象。
But you're saying twin prime is not an infinite monkey of phenomena. I mean, it's a very subtle monk it's still an infinite monkey phenomenon.
是的。如果质数真的是完全随机的，如果质数是由猴子生成的，那么，事实上，
Yeah. If the primes were really genuinely random, if the primes were generated by monkeys, then, yes, in fact,
无限猴子定理会...哦，但你是说孪生素数...你不能使用相同的工具。就像，它看起来几乎不是随机的。
the infinite monkey theorem would Oh, but you're saying that twin prime is it doesn't you can't use the same tools. Like, the it it doesn't appear random almost.
嗯，我们不知道。是的。我们相信素数表现得像一个随机集合。所以我们关心Trimheim猜想的原因是，它是一个测试案例，检验我们是否能真正自信地、零错误概率地说素数表现得像一个随机集合。好的。
Well, we don't know. Yeah. We we we we believe the primes behave like a random set. And so the reason why we care about the Trimheim conjecture is is is a test case for whether we can genuinely confidently say with with 0% chance of error that the primes behave like a random set. Okay.
随机，是的。我们知道素数的随机版本包含孪生素数，至少有100%的概率，或者随着范围扩大概率趋向100%。是的。所以我们相信素数是随机的。算术级数不可破坏的原因是，无论你的集合看起来是随机的还是有结构的（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。
Random yeah. Random versions of the primes we know contain twins, at least with with with 100% probability or probably tending to 100% as you go out further and further. Yeah. So the primes we believe that they're random. The reason why academic progressions are indestructible is that regardless of whether you're set it looks random or looks structured, like periodic, in both cases, ethnic progressions appear, but for different reasons.
这基本上就是所有方式...有很多关于这类算术级数定理的证明，它们都是通过某种二分法证明的：你的集合要么是有结构的，要么是随机的。在两种情况下你都能得出结论，然后把两者结合起来。但对于孪生素数，如果素数是随机的，那么你就赢了。但如果素数是有结构的，它们可能以某种特定方式结构化而消除孪生素数。我们无法排除那种阴谋论。
And this is basically all the ways in which the thing there are many proofs of of these sort of pathetic progression of theorems, 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. But in twin primes, if if the primes are random, then you're You win. But if your parents are structured, they can be structured in in a specific way that eliminates the the twins. And we can't rule out that one conspiracy.
然而据我理解，你们在k元组版本上取得了进展。
And yet you were able to make a, as I understand, progress on the k tuple version.
对的。是的。关于阴谋论的有趣之处在于，任何单一阴谋论都很难被驳倒。嗯。如果你相信世界是由蜥蜴人统治的，我出示证据证明不是，但你会说那证据是蜥蜴人伪造的。
Right. Yeah. So the the funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. Uh-huh. That, you know, if if you believe the word is run by lizards, you say here's some evidence that that it it not run by lizards, but that that evidence was planted by the lizards.
是的。对的。你可能遇到过这种现象。是的。所以，就像，几乎没有办法明确地...数学中也是如此，一个致力于研究孪生素数的学者会说...
Yeah. Right. You may have encountered this kind of phenomenon. Yeah. So so, like like, a pure like, there's there's almost no way to definitively without a and the same is true in mathematics that a consumer says taught solely devote devoted to learning twin primes.
你知道吗？就像，你还得渗透到数学的其他领域，但但，就像，它至少据我们所知是可以做到一致的。但有个奇怪的现象，你可以用一个阴谋论排除其他阴谋论。所以，如果世界是由蜥蜴人统治的，那也可能由外星人统治。对吧。
You know? Like, it would you would have to also infiltrate other areas of mathematics to sort of but but, like, you it could be made consistent, at least as far as we know. But there's a weird phenomenon that you can make one one conspiracy rule out other conspiracies. So, you know, if the if the world is is run by lizards, they can also be run by aliens. Right.
所以一个不合理的事情很难反驳，但超过一个，就有工具了。比如，我们知道有无穷多个素数对，它们之间的差最多为246，实际上这是目前的
So one unreasonable thing is is is is hard to dispute, but but more than one, there are there are tools. So, yeah, so for example, we we know there's infinitely many primes that are no no two which had so the infinitely pairs of primes which differ by at most 246 actually is is is the is the current
哦，所以有一个界限。是的。关于
Oh, so there's, like, a bound Yes. On the
对。就像，有孪生素数。还有叫表亲素数的，差为四。还有叫性感素数的，差为六。
Right. So, like, there's twin primes. There's this thing called cousin primes that differ by by four. There's thing called sexy primes that differ by six.
什么是
What are
性感素数？差为六的素数。这个名字——代价远没有名字听起来那么刺激。明白了。所以你可以用一个阴谋论排除其中一个，但一旦你有，比如，50个，结果是你无法同时排除所有。
sexy primes? Primes that differ by six. The the name the name is much less the cost is much less exciting than the name suggests. Got it. 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.
在这个阴谋空间里，这 somehow 需要太多能量。怎么
It just requires too much energy somehow in this conspiracy space. How do
你做了边界部分？你如何为不同的基因开发一个边界，使其获得奖励？好的。
you do the bound part? How do you how do you develop a bound for the different gene that prizes it Okay.
所以那个
So that
有无限多个？
there's an infinite number of?
所以它最终基于所谓的鸽巢原理。鸽巢原理是指，如果你有一定数量的鸽子，它们都必须进入鸽巢，而鸽子的数量多于鸽巢的数量，那么其中一个鸽巢必须至少容纳两只鸽子。所以必然会有两只鸽子靠得很近。例如，如果你有100个数字，它们都在1到1000的范围内，其中两个数字最多相差10。嗯。
So it's ultimately based on what's called the pigeonhole principle. So the pigeonhole principle is a statement that if you have a number of pigeons and they all have to go into into pigeonholes and you have more pigeons than pigeonholes, then one of the pigeon holes has to have at least two pigeons there. So there has to be two pigeons that are close together. So for instance, if you have a 100 numbers and they all range from one to a thousand, two of them have to be at most 10 apart. Mhmm.
因为你可以将1到100的数字分成100个鸽巢。假设你有101个数字。如果你有101个数字，那么其中两个数字的距离必须小于10，因为其中两个必须属于同一个鸽巢。所以这是数学中的一个基本原理。但它不能直接应用于质数，因为质数随着数值增大变得越来越稀疏。
Because you you you can divide up the numbers from one to a 100 into 100 pigeon holes. Let's let's say you have 100 you have a 101 numbers. If you have a 101 numbers, then two of them have to be distance less than 10 apart because two of them had to belong to the same pigeon hole. So it it's a basic basic feature of a basic principle in mathematics. So it doesn't quite work with the primes directly because the primes get sparser and sparser as you go out.
质数的数量越来越少。但事实证明，有一种方法可以给数字分配权重。比如有些数字几乎是质数，但它们并非完全没有除了自身和1以外的因子。它们有非常少的因子。事实证明，我们对几乎质数的理解比对质数的理解要好得多。
That that fewer and fewer numbers are prime. But it turns out that there's a way to assign weights to the to to numbers. Like so there are numbers that are kind of almost prime, but they're not they they don't have no factors at all other than themselves and one. They have very few factors. And it turns out that we understand almost primes a lot better than us than primes.
因此，例如，长期以来人们就知道他们在处理几乎质数。这方面已经有所研究。所以几乎质数是我们能够理解的东西。实际上你可以将注意力限制在一组合适的几乎质数上。虽然质数总体上非常稀疏，但相对于几乎质数来说，它们实际上稀疏程度要低得多。
And so, for example, it was known for a long time that they were treating almost primes. This has been worked out. So almost primes are something we we can't understand. So you can actually restrict attention to a a suitable set of almost primes. And whereas the primes are very sparse overall, relative to the almost primes, they actually are much less sparse.
他们让你可以建立一组几乎全是素数的集合，其中素数的密度约为1%。这样你就有机会通过应用某种原始原理来证明存在仅相隔100或更远的素数对。但为了证明修剪素数猜想，你需要让素数在整体中的密度达到50%的阈值。一旦达到50%，你就能得到修剪素数。但不幸的是，存在一些障碍。
They make you can set up a a set of almost primes where the primes have density, like, say, 1%. And that gives you a shot at proving by applying some sort original principle that that there's pairs of primes that are just only a 100 a 100 and above. But in order to with the trimmed prime conjecture, you need to get the density of primes inside the all sides up to up to a threshold of 50%. Once you get up to 50%, you would get trimmed primes. But, unfortunately, there are barriers.
我们知道，无论你选择哪种好的温芽集合，素数的密度永远无法超过50%。这被称为奇偶性障碍。我非常希望能找到突破这个障碍的方法，因为这将不仅打开修剪素数猜想的大门，还能解决哥德巴赫猜想以及数论中许多其他目前被阻碍的问题，因为我们现有的技术需要超越这个理论上的奇偶性障碍。这就像试图超越光速一样。
We know that that no matter what kind of good set of warm sprouts you pick, the density of primes can never get above 50%. It's called the parity barrier. And I would love to find yeah. So one of my long term dreams is to find a way to breach that barrier because it would open up not only to trim up conjecture, but the go back conjecture and many other problems in number theory are currently blocked because our current techniques would require improve going beyond this theoretical parity barriers. It's like it's like pulling past the speed of light.
是的。我们应该提到孪生素数猜想，这是数学史上最大的问题之一，还有哥德巴赫猜想。它们感觉就像是隔壁邻居。有没有那么几天，你觉得看到了解决问题的路径？
Yeah. So we should say a twin prime conjecture, one of the biggest problems in the history of mathematics, Golbak conjecture also. They feel like next door neighbors. Has there been days when you felt you saw the path?
哦，是的。有时候你尝试某种方法，效果特别好。这又回到了我们之前谈到的直觉。你从经验中学习，知道什么时候事情进展得太顺利了。
Oh, yeah. Yeah. Sometimes you try something, and it it works super well. You you again again, the sense of smell we talked about earlier. You learn from experience when things are going too well.
因为有些困难是你必须要遇到的。我想我的同事可能会这样形容：就像你在纽约街头蒙上眼睛坐进车里，几个小时后摘下眼罩，发现自己在北京。你知道我的意思吗？这 somehow 太容易了，好像没有穿越海洋。
Because there are certain difficulties that you sort of have to encounter. I think the way of of colleague might put it is that, you know, like, if if you are on the streets of New York and you put in a blindfold and you put in a car and and after some hours, you the blindfolds off, and then you're in Beijing. You know? I mean, that was too easy somehow. Like like, there was no ocean being crossed.
即使你不确切知道发生了什么，你也会怀疑有些事情不对劲。
Even if you don't know exactly what how what what was done, you're suspecting that that's something what wasn't right.
但这个想法是否还在你脑海中？你会不会时不时地回到素数问题上看看？
But is that still in the back of your head to do you return to these to the prime do you return to the prime numbers every once in a while to see?
是的。当我没什么更好的事情可做时——这种情况越来越少了，因为最近我有很多事情要忙。不过，是的，当我有空闲时间，又不想或者太沮丧而不愿处理真正的研究项目时，我也不想做行政事务或为家人跑腿。我就会玩玩这些东西找乐子，通常都是毫无进展。
Yeah. When I have nothing better to do, which is less and less than that, is I get busy with so many things these days. But, yeah, when I have free time and I'm not and I'm too frustrated to to work on my sort of real research projects. And I also don't want to do my administrative stuff or I don't want to do some errand for my family. I can play with these these things for fun, and usually, you get nowhere.
是的。你得学会坦然接受，好吧，又一次一无所获。我会继续前进。
Yeah. You have to learn to just say, okay. Fine. I once again, nothing happened. I will I will move on.
是的。极少数情况下，我确实解决了其中某个问题，或者有时候如你所说，你以为解决了，然后兴奋个大概十五分钟，接着想我应该检查一下，因为这太简单太美好了，不像是真的——通常确实不是。
Yeah. Very occasionally, one of these problems I actually solved or sometimes, as you say, you think you solved it, and then you're euphoric for maybe fifteen minutes, and then you think I should check this because this is too easy too good to be true, it usually is.
你对这些问题何时能解决有什么直觉？比如孪生素数猜想，回头说说这个。
What's your gut say about when these problems would be solved? Put in Prime and go back.
孪生素数猜想，我认为我们会不断获得更多部分结果。它至少需要突破这个奇偶性障碍，这是目前最大的遗留难题。这个猜想有一些更简单的版本，我们已经非常接近解决了。所以我认为十年内，我们会得到更多、更接近的结果，但可能还无法完全解决整个问题。
Prime, I think we will keep getting keep getting more partial results. It does need at least one this parity barrier is is the biggest remaining obstacle. There are simpler versions of the conjecture where we are getting really close. So I think we will in ten years, we will have many more much closer results. We may not have the whole thing.
是的。所以趋势上是在逐步接近。嗯。黎曼假设，我没什么头绪——我的意思是，它恐怕得靠偶然发现才能解决。
Yeah. So trend times is somewhat close. Yep. Riemann hypothesis, I have no I mean, it has to happen by accident.
我认为黎曼假设是一种关于素数分布更广义的猜想，对吧？
I think the Riemann hypothesis is is a kind of more general conjecture about the distribution of prime numbers. Right?
是的。从应用角度来理解是比较安全的。比如，对于只涉及乘法而不涉及加法的问题，质数的行为确实如你所期望的那样随机。在概率论中有一个称为平方根抵消的现象，就像如果你想就某个问题对美国进行民意调查，如果你只询问一两个选民，可能抽样偏差很大，那么你对整体平均值的测量就会非常不精确。但如果你抽样的人越来越多，准确性就会越来越高，实际上它的改善程度与你抽样人数的平方根成正比。
Yeah. It's it's it's safe to sort of viewed more applicatively. Like, for for questions only involving multiplication, no addition, the primes really do behave as randomly as as you could hope. So there's a phenomenon in probably called scroger cancellation that, you know, like, if if you wanna poll, say, America upon some issue, and you you ask one or two voters, you may have sampled a bad sample, and then you get you get a really imprecise measurement of of a full average. But if you sample more and more people, the accuracy gets better and better, and it actually improves like the square root of the number of of people you you sample.
所以，如果你抽样一千人，你可以得到大约3%的误差范围。同样地，如果你在某种乘法意义上测量质数，有一种特定的统计量可以测量，它被称为函数。它会上下波动，但在某种意义上，随着你不断进行更多平均，如果你抽样越来越多，波动应该会下降，就像它们是随机的一样。有一种非常精确的方法来量化这一点，而这个假设正是以非常优雅的方式捕捉了这种特性。
So, yeah, if you sample a thousand people, you can get, like, a 3% margin of error. So in the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you you can measure, and it's it's called the 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 hypothesis is a very elegant way that captures this.
但就像数学中的许多其他领域一样，我们很少有工具能够证明某物真正表现得像完全随机。这个部分不仅仅是有点随机，而是要求它的行为像一个真正随机集合那样随机，这种平方根抵消现象。而且我们知道，由于与奇偶性问题相关的原因，我们大多数常规技术都无法解决这个问题。证明必须来自意想不到的方向。是的。
But as with many other ways in mathematics, we have very few tools to show that something really genuinely behaves like, really random. And this section is not just a little bit random, but it's it's asking that it behaves as random as an actually random set, this this this square root cancellation. And we know because of things related to parity problem actually that most of us usual techniques cannot hope to settle this question. The proof has to come out of left field. Yeah.
但那具体是什么，是的。没有人有任何严肃的提议。是的。而且，正如我所说，有各种方法可以稍微修改质数，这样你就会破坏这个假设。所以，它必须非常微妙。
But what that is yeah. No one has any serious proposal. Yeah. And and there's there's various ways to sort of as I said, you you can modify the primes a little bit, and you can destroy the hypothesis. So, like, it has to be very delicate.
你不能应用那些误差范围很大的方法。它必须刚好有效。而且，就像有很多陷阱，你需要非常巧妙地避开。
You can't apply something that has huge margins of error. It has to just barely work. And, like, there's, like, all these pits pitfalls that you, like, dodge very adeptly.
是的。质数真是令人着迷。
Yeah. The prime numbers is just fascinating.
是的。是的。是的。
Yeah. Yeah. Yeah.
对你来说，素数最神秘的地方是什么？
What what to you is most mysterious about the prime numbers?
这是个好问题。从猜想的角度来看，我们有一个很好的模型。就像我说的，它们确实有一些规律。比如，素数通常是奇数。但除了这些显而易见的规律外，它们的行为非常随机。
So that's a good question. So so, like, conjecturally, we have a good model of them. I mean, like, 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.
假设它们的行为符合所谓的克拉默素数随机模型，即在某个点之后，素数就像随机集合一样行为。这个模型有一些细微的调整，但它一直是个非常好的模型。它与数值匹配，告诉我们该预测什么。比如，我可以完全确定地告诉你真相。
And just assuming that they behave so there's something called the Cramer random model of the primes that that that after a certain point, primes just behave like a random set. And there's various slight modifications to this model, but this has been a very good model. It matches the numerics. It tells us what to predict. Like, I can tell you with complete certainty the true.
随机模型给出了压倒性的概率表明它是真的。我只是无法证明它。我们的大部分数学都是为了解决有规律的事物而优化的，而素数却具有这种反规律性，几乎所有事物都是如此。但我们无法证明这一点。是的。
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 primes have this anti pattern, as do almost everything, really. But we can't prove that. Yeah.
我想素数表现得随机并不神秘，因为它们没有理由拥有任何秘密规律。但神秘的是，究竟是什么机制真正迫使这种随机性发生？而这正是我们所缺失的。
I guess it's not mysterious that the primes being ran 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 absent.
另一个令人惊讶的难题是考拉兹猜想。哦，是的。表述简单，嗯。可视化很美，是的。它简单却极难解决，然而你已经取得了一些进展。
Another incredibly surprisingly difficult problem is the Collatz conjecture. Oh, yes. Simple to state Mhmm. Beautiful to visualize Yes. In its simplicity, and yet extremely difficult to solve, and yet you have been able to make progress.
保罗·埃尔德什谈到考拉兹猜想时说，数学可能还没准备好解决这样的问题。其他人也说过这是一个异常困难的问题，完全超出了当前数学的能力范围。这是在2010年。超出了当今数学的触及范围，然而，是的，你已经取得了一些进展。
Paul Ridar said about the Colas 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 Yeah. You have made some progress.
为什么这么难做？你能解释清楚它到底是什么吗？哦，是的。是的。
Why is it so difficult to make? Can you actually even explain what it is? It's Oh, yeah. Yeah.
所以这是一个可以解释的问题。是的。它需要一些视觉辅助，但确实如此。你取任意自然数，比如13，然后对它应用以下程序：如果是偶数，就除以二。
So it's it's a it's a problem that you can explain. Yeah. It it helps with some visual aids, but yeah. So you take any natural number, like, say, 13, and you apply the the following procedure to it. So if it's even, you divide it by two.
如果是奇数，就乘以三再加一。所以偶数会变小，奇数会变大。13会变成40，因为13乘以三是39，加一就是40。这是一个简单的过程。
And if it's odd, you multiply it by three and add one. So even numbers get smaller, odd numbers get bigger. So 13 will become 40 because 13 times three is 39. Add one to your 40. So it's a simple process.
对于奇数和偶数来说，都是非常简单的操作。组合起来也还算简单。但当你问迭代会发生什么时，你把你得到的输出再输入回去。所以13变成40。
For odd numbers and even numbers, they're both very easy operations. And then you put together, it's still reasonably simple. But then you you ask what happens when you iterate it. You you take the output that you just got and feed it back in. So 13 becomes 40.
40现在是偶数，除以二是20。20还是偶数，除以二是10。10除以二是5，然后5乘以三加一是16，然后是8、4、2、1。
40 is now even. Divide by two is 20. 20 is still even. Divide by 10 to 10. Five and then five times three plus one is 16, and then eight four two one.
然后从1开始，它会循环1421421，永远循环下去。我刚才描述的序列，13、40、20、10等等，这些也被称为冰雹序列，因为有一个过度简化的冰雹形成模型（虽然实际上不太准确），但不知为何被教给高中生作为初步近似：就像一小块冰核在云中形成冰晶，由于风的作用上下移动，有时在寒冷时获得更多质量，可能融化一点。这种上下运动的过程形成了部分融化的冰，最终形成冰雹，最后落到地面。
So and then from one, it goes 1421421. It cycles forever. So the sequence I just described, you know, thirteen, forty, twenty, ten, so forth, these are also called hailstorm sequences because there's an oversimplified model of of hailstorm formation, yeah, which is not actually quite correct, but it's so somehow taught to high school students as a first approximation is that, like a a little nugget of ice gets gets an ice crystal forms in in a cloud, and it goes up and down because of the wind. And and sometimes when it's cold, it gets it acquires a bit a bit more mass, and maybe it melts a little bit. And this process of going up and down creates this sort of partially melted ice, which eventually gives us hailstorm, and eventually it falls out the earth.
所以猜想是，无论你从多高的数字开始，比如百万或十亿级别的数，经过这个奇数上升、偶数下降的过程，最终都会降到地面
So the conjecture is that no matter how high you start up, like, take a number which is in the millions or billions, you go this process that that goes up if you're odd and and down if you're even, eventually goes down to to to earth
总是如此。无论你从这个非常简单的算法从何处开始，最终都会到达1。没错。而且你可能会攀升一段时间
all the time. No matter where you start with this very simple algorithm, end up at one. Right. And you might climb for a while
没错。是的。所以这是未知的。是的。如果你绘制这些序列，它们看起来像布朗运动。
Right. Yeah. So it's unknown. Yeah. If you plot it, these sequences, they look like Brownian motion.
它们看起来像股市。你知道，它们只是以一种看似随机的模式上下波动。事实上，通常情况就是这样，如果你输入一个随机数，你至少可以初步证明它看起来像随机游走。而且这实际上是一种带有向下漂移的随机游走。就像你总是在赌场玩轮盘赌，而赔率略微对你不利。
They look like the stock market. You know, they just go up and down in a in a seemingly random pattern. And in fact, usually, that's what happens, that that if you plug in a random number, you can actually prove at least initially that it would look like random walk. And that's actually a random walk with a downward drift. It's like if you're always gambling on on a roulette at at the casino with odds slightly weighted against you.
所以有时你赢，有时你输。但从长远来看，你输的比赢的多一点。所以通常情况下，如果你一直不停地玩下去，你的钱包最终会归零。
So sometimes you 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 hit will go to zero if you just keep playing over and over again.
所以从统计上看，这是有道理的。
So statistically, it makes sense.
是的。所以我证明的结果大致是这样的：从统计上看，大约99%的输入都会向下漂移——可能不会一路降到1，但会比开始时小得多。就像我告诉你，如果你去赌场，大多数时候，只要你玩得足够久，最终你钱包里的钱会比开始时少。这就是我证明的结果。
Yes. So so the result that I I proved roughly speaking is such that that statistically, like, 99% of all inputs would would drift down to maybe not all the way to one, but to be much, much smaller than what you started. So it's 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 it for long enough, you end up with a smaller amount of in your water than when you started. That's kind of like the what the result that I proved.
那么这个结果为什么……你能沿着这个思路继续证明整个猜想吗？
So why is that result like, can you continue down that thread to prove the full conjecture?
嗯，问题是，我使用了概率论中的论点，而概率论中总是存在这种例外事件。所以，在概率中，我们有这些大数定律，它告诉你，如果你在赌场玩一个期望值为负的游戏，随着时间的推移，你几乎必然——概率可以任意接近100%——会输钱。但总是存在这种异常的离群值。就像，在数学上是可能的，即使游戏赔率对你不利，你仍然可能赢的次数略多于输的次数。这很像纳维-斯托克斯方程中可能出现的情况，大多数时候，你的波可能会消散。
Well, the the problem is that my I I used arguments from probability theory, and there's always this exceptional event. So, you know, so in probability, we have these these low large numbers, which tells you things like if you play a casino with a a game at a casino with a losing expectation, over time, you are guaranteed well, almost surely, with probably probably as close to 100 as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. Like, it is mathematically possible that even in in the game is is the odds are not in your favor, 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.
可能只有一个离群的初始条件选择会导致你爆炸，也可能有一个特殊的离群数字被插入，它会冲向无穷大，而所有其他数字都会坠落到地面，坠落到一。是的。事实上，有一些数学家，比如亚历克斯·康托罗维奇，提出这些考拉兹迭代实际上类似于这些类似的自动机。实际上，如果你观察二进制中的情况，它们确实看起来有点像这些生命游戏类型的模式。类比生命游戏如何创造这些巨大的自我复制对象等等，可能你可以创造某种比空气重的飞行机器，一个实际上编码这种机器的数字，它的任务就是编码并创造一个更大的自身版本。
There could be just one outlier choice of initial conditions that would lead you to blow up, and there could be one outlier choice of special number that they stick in that shoots off infinity while all other numbers crash to earth crash to one. Yeah. In fact, there's some mathematicians who've Alex Kontorovich, for instance, who've proposed that that actually these collats iterations are like these similar automata. Actually, if if you look about the happen on on in binary, they do actually look a little bit like like these Game of Life type patterns. And in an analogy to how the Game of Life can create these these massive, like, self replicating 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 itself which is which is larger.
编码在数字中的比空气重的机器，是的。永远飞行。
Heavier than air machine encoded in a number Yeah. That flies forever.
是的。事实上，康威也研究过这个问题。哦，哇。所以康威实际上很相似，那更多是我们纳维-斯托克斯项目的灵感来源。康威研究了考拉兹问题的推广，其中不是通过乘以三加一或除以二来变形，而是有一组更复杂的规则。
Yeah. So Conway, in fact, worked on worked on this problem as well. Oh, wow. So Conway so similar in fact, that was more of our inspirations for the Navi Navi Stokes project. Conway studied generalizations of the collapse problem where instead of morphing by three and adding one or dividing by two, you you have a more complicated bunch of words.
但不是有两种情况，也许你有17种情况，然后你上下波动。他表明，一旦你的迭代变得足够复杂，你实际上可以编码图灵机，并且可以使这些问题不可判定，并做类似这样的事情。事实上，他为这类分式线性变换发明了一种编程语言。他称之为Factran，是对Fortran的一种戏仿，并且表明它是图灵完备的，你可以编程。你可以制作一个程序，如果你插入的数字被编码为质数，它就会同步到零。
But 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 Turing machines, and you can actually make these problems undecidable and and do things like this. In fact, he invented programming language for these kind of fractional linear transformations. He called a fact track as a play on on Fortran, and he showed that that you you couldn't you you can program as it was Turing complete. You could you could you could could make a program that if if your number you insert in was encoded as a prime, it would sync to zero.
它会下降。否则，它会上升等等。所以这类普遍的问题确实和所有数学一样复杂。
It would go down. Otherwise, it would go up and things like that. So the general class of problems is is really as complicated as all the mathematics.
我们讨论过的细胞自动机的某些奥秘，要有一个数学框架来谈论细胞自动机，可能需要同样类型的框架。
Some of the mystery of the cellular automata that we talked about having a mathematical framework to say anything about cellular automata may be the same kind of framework is required.
是的。是的。Glox注入器。是的。如果你想做的不是统计意义上的，而是真正想要百分之百、百分之百的所有输入都用于地球。
Yeah. Yeah. Glox injector. Yeah. If you want to do it not statistically, but you really want one hundred one hundred percent of all inputs to to for the earth.
是的。所以可能可行的是，嗯，69%，你知道，朝着一个目标努力，但要覆盖所有方面，你知道，那看起来很难。
Yeah. So what might be feasible is is, yeah, 69%, you know, go go to one, but to like, everything, you know, that looks hard.
在这些著名问题中，你认为哪个是目前我们面临的最难题？是黎曼猜想吗？
What would you say is out of these within reach, famous problems is the hardest problem we have today? Is it Riemann hypothesis?
黎曼猜想名列前茅。P对NP问题（PCOSP）也是一个很好的例子，因为，就像，那是一个元问题。如果你在积极意义上解决了它，即找到了一个P对NP算法，那么这可能也会解决许多其他问题。我们应该提一下我们一直在讨论的一些猜想。
Riemann is up there. PCOSP is a good one because, like, that's that's that's a meta problem. Like, if you solve that in the in the positive sense that you can find a PCOSP algorithm, then potentially, this solves a lot of other problems as well. And we should mention some of the conjectures we've been talking about.
你知道，现在很多东西都建立在它们之上。有连锁反应。P对NP问题（P vs NP）的连锁效应基本上比任何其他问题都大。对的。
You know, a lot of stuff is built on top of them now. There's ripple effects. P goes on p has more ripple effects than basically any other Right.
如果黎曼猜想被证伪，那将对数论学家是一个巨大的心理冲击，但它也会对密码学产生后续影响。
If the Riemann hypothesis is disproven, that'd be a big mental shock to a number theorist, but it would have follow on effects for cryptography.
嗯哼。
Uh-huh.
因为许多密码学都运用数论中涉及素数等的构造方法。它非常依赖于数论学家多年来建立的直觉，即哪些涉及素数的运算表现得随机，哪些不随机。特别是，我们的加密方法旨在将含有信息的文本转化为与随机噪声无法区分的文本。因此，我们相信这几乎不可能被破解，至少在数学上是如此。但如果我们的信念——我们的假设——是错误的，那就意味着存在我们尚未意识到的素数实际模式。
Because a lot of cryptography uses number theory uses number theory constructions involving primes and so forth. And it relies very much on the intuition that number theorists have built over many, many years of what operations involving primes behave randomly and what ones don't. And in particular, our encryption methods are designed to turn text with information on it into text which is indistinguishable from from random noise. So and hence, we believe to be almost impossible to crack, at least mathematically. But if something has caught our beliefs our hypothesis is is wrong, it means that there are there are actual patterns of the primes that we're not aware of.
而如果存在一个，很可能还有更多。突然间，我们的许多加密系统都受到质疑。是的。
And if there's one, there's probably going more. And suddenly, lot of our crypto systems are in doubt. Yeah.
但那你又如何谈论素数呢？是的。你现在又朝着考拉兹猜想的方向去了。因为如果我...你希望它是随机的，对吧？
But then how do you then say stuff about the the primes? Yeah. Now you're going towards the coax conjecture again. Because if I I I you do you want it to be random. Right?
你希望它是随机的。
You want it to be random.
所以更广泛地说，我只是在寻找更多工具、更多方法来证明...是的，证明事物是随机的。你如何证明阴谋不会发生？对吧。
So more broadly, I'm just looking for more tools, more ways to show that that Yeah. That things are random. How do you prove a conspiracy doesn't happen? Right.
对你来说，P=NP有任何可能性吗？你能想象出一个可能的宇宙吗？这是可能的。
Is there any chance to you that p equals NP? Is there some can you imagine a possible universe? It is possible.
我的意思是，存在各种情景。从技术上讲，有一种方式是可能的，但实际上永远无法实现。证据稍微倾向于否定，即我们可能P不等于NP。
I mean, there's there's various scenarios. I mean, there's 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 a good NP.
我的意思是，这看起来更像是黎曼猜想那种情况。我认为证据相当有力地倾向于否定的一面。
I mean, it seems like it's one of those cases seem more similar to Riemann hypothesis. It I think the evidence is leaning pretty heavily on the no.
当然更倾向于否定而非肯定。关于P与NP问题的有趣之处在于，我们面临的障碍比几乎任何其他问题都要多得多。所以虽然有证据支持，但我们也有很多结果排除了许多许多解决问题的方法。这是计算机科学实际上非常擅长的一点——明确指出某些方法行不通。
Certainly more on the no than on the on the yes. The funny thing about PCOS and p 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 has actually been very good at. It's actually saying that that certain approaches cannot work.
不可行定理。它可能是不可判定的。我们不知道，是的，我们不知道。
No go theorems. It could be undecidable. We don't yeah. We don't know.
我读到一个有趣的故事：当你获得菲尔兹奖时，有网友写信问你，赢得了这个 prestigious 奖项后打算做什么？然后你很快速、很谦虚地回答，说这块闪亮的金属并不能解决我正在研究的任何问题。
There's a funny story I read that when you won the Fields Medal, somebody from the Internet wrote you and asked, you know, what are you gonna do now that you've won this prestigious award? And then you just quickly, very humbly said that, you know, this shiny metal is not gonna solve any of the problems I'm currently working on.
所以就
So just
我会继续研究它们。首先我觉得很有趣的是你会在那种情况下回复邮件。其次，这显示了你的谦逊。不过，也许你可以谈谈菲尔兹奖，但这也是我问起格里戈里·佩雷尔曼的另一种方式。你怎么看他 famously 拒绝接受菲尔兹奖和与之伴随的百万美元千禧年奖金的决定？
I'm gonna keep I'm gonna 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 it just shows your humility. But, anyway, maybe you could speak to the Fields Medal, but it's another way for me to ask about, Gregorio Perlman. What do you think about him famously declining the Fields Medal in the millennial prize, which came with a $1,000,000 of prize money?
他表示：'我对金钱或名誉不感兴趣。这个奖对我来说完全无关紧要。如果证明是正确的，那么不需要任何其他认可。'
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.
是的。不。他算是比较特立独行的人，即使在通常持有理想主义观点的数学家中也是如此。我从未见过他。我想有一天会有兴趣见见他，但我一直没有机会。
Yeah. No. He's he's somewhat of an outlier, even among mathematicians who tend to to have somewhat idealistic views. I've never met him. I think I'd be interested to meet him one day, but I I never had the chance.
我认识一些见过他的人。他对某些事情总是有强烈的看法。你知道吗？我的意思是，他并非完全与数学界隔绝。他会做演讲、写论文等等。
I I know people who met him. He's always had strong views about certain things. You know? I mean, it's it's not like he was completely isolated from the the math community. I mean, he would he would give talks and and write papers and so forth.
但在某个时候，他决定不再与社区其他人互动。他可能是幻灭了或什么的。我不知道。他决定退出，在圣彼得堡采蘑菇之类的，这也没什么。你知道吗？
But at some point, just decided not to engage with the rest of the community. He was he was disillusioned or something. I don't know. And he decided to to peace out and, you know, collect mushrooms in Saint Petersburg or something, and and that's that's fine. You know?
你可以这么做。我的意思是，这是另一面。我们解决的许多问题中，有些确实有实际应用，这很好。但如果你停止思考某个问题，你知道，他之后就没在这个领域发表过论文，但这没关系。也有很多人这样做过。
And you can you can do that. 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, and that's that's great. But, like, if you stop thinking about a problem, you you know, so he's he hasn't published since in this field, but that's fine. There's many, other people who've done so as well.
是的。所以我想我最初没有意识到菲尔兹奖的一点是，它某种程度上让你成为体制的一部分。嗯。你知道吗？大多数数学家只是职业数学家。
Yeah. So I guess one thing I didn't realize initially with the Fields Medal is that it it sort of makes you part of the establishment. Mhmm. You know? So, you know, most mathematicians, you know, there's they're just career mathematicians.
你知道吗？你只专注于发表下一篇论文，也许通过一次评审晋升一级，开始一些项目，带几个学生之类的。但突然之间，人们想要你的意见，你必须多思考一下，因为你可能会轻率地说些话，但现在更重要了，因为有人会听。
You know? You just focus on publishing the next paper, maybe getting one test to promote one one rank, you know, and and starting a few projects, maybe taking some students or something. Yeah. But then suddenly people want your opinion on things, and you have to think a little bit about the, you know, things that you might just so foolishly say because you know no one's gonna listen to you. It's it's more important now.
这对你来说是一种约束吗？你还能享受乐趣、做个叛逆者、尝试疯狂的事情吗？
Is it constraining to you? Are you able to still have fun and be a rebel and try crazy stuff and
玩转想法？我现在比以前空闲时间少多了。主要是自己选择的。我的意思是，我显然可以选择拒绝，所以我拒绝了很多事情。
Well play with ideas? I have a lot less free time than I had previously. I mean, mostly by choice. I mean, I I I can obviously, I have the option to sort of decline. So I decline a lot of things.
我本可以拒绝更多。或者我可以变得如此不可靠，以至于人们不再来问我。但我喜欢这里不同的算法。
I I I could decline even more. Or I could acquire a reputation for being so unreliable that people don't even ask anymore. But this is I love the different algorithms here.
这太棒了。
This is great.
这始终是一个选择。但你知道，有些事情，就像我说的，我不像做博士后时那样花那么多时间一次只钻研一个问题或者随便玩玩。我仍然会做一些，但随着职业发展，一些软技能变得更重要。数学在某种程度上把技术技能都集中在了职业生涯早期。所以作为博士后，就是发表或淘汰。你需要专注于证明技术性定理来证明自己。
This is it's it's always an option. But, you know, there are things that are, you know, like I mean, so I mean, I I I don't spend as much time as I do as a postdoc, know, just just working at one problem at a time or fooling around. I still do that a little bit, but, yeah, as you advance in your career, some of the more soft skills so math somehow front loads all the technical skills to the early stages of your career. So, yeah, so it's a as a postdoc, it's published or perish. You you you incentivize to basically focus on on proving very technical theorems to sort prove yourself as well as prove the theorems.
但随着资历加深，你开始需要指导他人、接受采访、尝试塑造领域方向——无论是研究方面还是有时需要处理各种行政事务。这是一种合理的社会契约，因为你需要在一线工作才能了解什么能真正帮助数学家。
But then as as you get more senior, you have to start, you know, mentoring and and and and giving interviews and and trying to shape direction of the field both research wise and and, you know, sometimes you have to, you know, do various administrative things. And it's kind of the right social contract because you you need to to work in the trenches to see what can help mathematicians.
体制的另一面，真正积极的一点是，你可以成为激励许多年轻数学家或对数学感兴趣的年轻人的灯塔。就像...是的。这就是人类思维的运作方式。这里我可能要说说我喜欢菲尔兹奖，因为它确实以某种方式激励了很多年轻人。
The other side of the establishment sort of the the really positive thing is that you get to be a light that's an inspiration to a lot of young mathematicians or young people that are just interested in mathematics. It's like Yeah. Yeah. It's just how the human mind works. This is where I would probably say that I like the Fields Medal, that it does inspire a lot of young people somehow.
我觉得这就是人脑的运作方式。是的。同时，我也想要尊重像格里戈里·佩雷尔曼这样的人，他在内心批判奖项。那是他的原则，任何人能够为了原则做到大多数人都做不到的事情，看到这样的事都很美好。
I don't this is just how human brains work. Yeah. At the same time, I also wanna give sort of respect to somebody like Gagorje Perelman 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 that most humans would not be able to do. It's beautiful to see.
某种程度的认可确实重要，但同样重要的是不要让这些事情主宰你的生活。是的。而且，不要只关心获得下一个大奖之类的。我的意思是，确实如此。所以，你会看到这些人试图只解决那些非常大的数学问题，而不去研究那些不那么吸引人但实际仍然有趣且有启发性的课题。
Some recognition is is necessarily important, but, yeah, it's it's also important to not let these things take over your life. Yeah. And, like, only be concerned about getting the next big award or whatever. I mean, it yeah. So, again, you see these people try to only solve, like, a really big math problems and and not work on on things that are less sexy if you wish, but but but actually still interesting and and is instructive.
正如你所说，人类思维运作的方式是，当事情与人相关联时，我们理解得更好。而且，如果只与少数人关联，就像我说的，我们人类思维的构造方式使我们能够理解10到20个人之间的关系。但一旦超过这个数量，比如100人，就存在一个极限。我认为这有个专业名称，超过这个数量后，其他人就变成了'他者'。是的。
As you say, like, the way the human mind works, it's we understand things better when they're attached to humans. And, also, if they're attached to a small number of humans, like I said, there's there's the way our humans mind is is wired, we can comprehend the relationships between 10 or 20 people. But once you get beyond that, like, 100 people, like, there's there's a there's a limit. I think there's a name for it beyond which it just becomes the other. Yeah.
因此你必须简化整个人类群体，你知道，99.9%的人类都变成了'他者'。而这些模型往往是不准确的，这会导致各种问题。所以，要将一个学科人性化，如果你确定一小群人，比如说他们是该学科的代表人物，比如榜样，这有一定作用，但过多也可能有害，因为我要第一个承认，我自己的职业轨迹并不典型。
And so we have you have to simplify the whole mass of, you know, 99.9% of humanity becomes the other. And often these models are are are incorrect, and this causes all kinds of problems. But so yeah. So to humanize a subject, you know, if you identify a small number of people, let's say, you these are representative people of the subject, know, role models, for example, That has some role, but it can also be yeah. Too much of it can be harmful because it's I'll be the first to say that my own career trap trap is not that of a typical mathematician.
我接受了非常加速的教育，跳过了很多课程。我认为我有非常幸运的导师机会，而且我觉得我在正确的时间出现在了正确的地方。仅仅因为某人没有我的发展轨迹，并不意味着他们不能成为优秀的数学家。我的意思是，他们可能以非常不同的风格做数学，我们需要不同风格的人。而且，你知道，有时候过多关注的是在数学或其他领域完成项目的最后一步的人，而实际上这些项目是经过数百年或数十年、大量前期工作积累才得以完成的。
I the very accelerated education, I skipped a lot of classes. I think I was had very fortunate mentoring opportunities, and I think I was at the right place at the right time. Just because someone does doesn't have my trajectory, you know, doesn't mean that they they can't be good mathematicians. I mean, they may be mathematicians in a very different style, and we need people with a different style. And, you know, even if and sometimes too much focus is given on the on the person who took the last step to complete a project in mathematics or elsewhere that's really taken, you know, centuries or decades with lots and lots of building a lots of previous work.
但如果你不是专家，这个故事很难讲述，因为你知道，只说一个人做了某件事要容易得多。明白吗？这样历史就简单多了。
But that's a a story that's difficult to tell if you're not an expert because, you know, it's easier to just say one person did this one thing. You know? It makes for a much simpler history.
我认为总体而言，将史蒂夫·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。嗯。我个人知道，当然每个人都知道那些令人难以置信的设计和工程团队
I think on the whole, it is a hugely positive thing to to talk about Steve Jobs Mhmm. As a representative of Apple When I personally know, and of course, everybody knows the incredible design, the incredible engineering teams
嗯。
Mhmm.
只是那些团队中的个体成员。他们不是一个团队，而是团队中的独立个体，那里有很多才华横溢的人。但这只是一个很好的简称，就像非常像派一样。是的。
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.
史蒂夫·乔布斯。
Steve Jobs.
是的。是的。派。作为一个起点，你看，作为第一个近似值。就是这样
Yeah. Yeah. Pie. As as a starting point, see, as a as a first approximation. That's how
你然后读一些传记，再深入了解更深层次的第一近似值。是的。没错。所以你提到你在普林斯顿大学时
you And then read some biographies, and then look into much deeper first approximation. Yeah. That's right. So you mentioned you were at Princeton to
嗯。
Mhmm.
安德鲁·怀尔斯当时在那里。
Andrew Wiles at that time.
哦，是的。他是那里的教授。嗯。
Oh, yeah. He's a professor there. Mhmm.
历史如何相互关联真是个有趣的时刻。当时他宣布去年从我们这里证明了它。现在有了更多背景，你对数学史上那个时刻有什么看法？是的，我当时
It's a funny moment how history is just all interconnected. And at that time, he announced that he proved it from us last year. What did you think maybe looking back now with more context about that moment in math history? Yes. I was
还是个研究生。我隐约记得，当时有媒体关注，我们都在同一个邮件室有信箱。大家都在取邮件时，突然安德鲁·怀尔斯的信箱爆满到溢出来。这是个很好的衡量标准。
a graduate student at the time. I mean, I I vaguely remember, you know, there was press attention, and we all had the same we had pigeonholes in the same mailroom. You know? So we were all picking up mail, and it was like suddenly Andrew Wiles' mailbox exploded to be overflowing. That's a good that's a good metric.
是的。所以我们在茶歇时都会讨论这个。我的意思是，我们大多数人并不真正理解证明，只了解一些高层次细节。
Yeah. You know? So, yeah, we we all talked about it at at tea and so forth. I mean, we we didn't understand most of us sort understand the proof. We understand sort of high level details.
事实上，现在有一个用Lean形式化验证它的持续项目。凯文·邦塞特正在做这个。
In fact, there's an ongoing project to formalize it in lean. Right? Kevin Ponset is actually.
是的。我们能稍微聊聊这个吗？这有多难？因为据我理解，费马大定理的证明涉及超级复杂的对象。
Yeah. Can can we take that small tangent? Is it is it how difficult is that? Because as as I understand the Fermat's last the the proof for Fermat's last theorem has, super complicated objects. Yeah.
现在要形式化验证确实非常困难。
That's really difficult to formalize now.
是的。我想你说得对。他们使用的那些对象是可以定义的，所以已经在Lean中定义了它们。
Yeah. I guess yeah. You're right. The the objects that they use, you can define them. So they've been defined in lean.
好的。所以仅仅定义它们是什么是可以做到的。这确实不简单，但已经有人做到了。但是关于这些对象有很多非常基本的事实，需要花费数十年时间才能证明，并且在所有这些不同的数学论文中都有涉及。因此，其中许多也需要被形式化。
Okay. So so just defining what they are can be done. That's really not trivial, but it's been done. But there's a lot of really basic facts about these objects that have taken decades to prove and that that that in all these different math papers. And so lots of these have to be formalized as well.
凯文·巴扎德的目标，实际上，他有一个为期五年的资助项目来形式化费马大定理。他的目标是，他认为自己无法一直追溯到基本公理，但他希望将其形式化到只需要依赖1980年数论学者已知的黑箱概念的程度。然后需要其他人或其他工作来从此处继续推进。所以这与我所熟悉的数学领域不同。在分析学，也就是我的领域，我们研究的对象更接近基础层面。
Kevin's Kevin Buzzard's goal, actually, he has a five year grant to formalize Fermi's last theorem. And his aim is that he doesn't think he will 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 to number theorists at the time. And then some other person or some other work would have to done to to to get from there. So it's it's a different area of mathematics than the type of mathematics I'm used to. In analysis, which is kind of my area, the objects we study are kind of much closer to the ground.
我研究诸如质数、函数等事物，这些至少在高中的数学教育范围内是可以定义的。是的。但数论还有这个非常高级的代数方面，人们在那里已经构建了层层结构相当长一段时间。这是一个非常稳固的结构。它的基础部分已经发展得非常完善，有教科书等等。
We study I study things like prime numbers and and and functions and and things that are within scope of a high school math education to at least define. Yeah. But then there's this very advanced algebraic side of number theory where people have been building structures upon structures for for quite a while. And it's it's a very sturdy structure. There's a it's it's been it's been very at the base at Middle East is extremely well developed with textbooks and so forth.
但这确实到了这样一个地步：如果你没有经过这些年的学习，而想询问这座塔第六层左右发生了什么，你需要花费相当多的时间才能甚至到达
But it does get to the point where if you're if you haven't taken these years of study and you wanna ask about what what is going on at, like, Level 6 of of this tower, you have to spend quite a bit of time before they can even get to the
那个你能看到你认识的人的程度。关于他这段旅程，是什么激励了你？正如我们谈到的，他七年 mostly 秘密工作，是否有相似之处？
point where you can see you see someone you recognize. What inspires you about his journey that was similar as we talked about seven years mostly working in secret?
是的。没错。那是一种浪漫，是的。所以它某种程度上符合我认为人们对数学家的那种浪漫想象——如果他们确实这么想的话——就是那种古怪的，你知道，像巫师之类的人。所以这某种程度上强化了那种视角。
Yeah. Yes. That is a a romantic yeah. So it kind of fits with the sort of the the romantic image I think people have of mathematicians to the extent they think of that at all as these kind of eccentric, you know, wizards or something. So that's something kind of accentuated that perspective.
你知道吗？我的意思是，这是一个伟大的成就。他解决问题的风格与我自己的如此不同，但这很棒。我的意思是，我们需要那样的人。
You know? I mean, it's it is a great achievement. His style of solving problems is so different from my own, but which is great. I mean, we we need people like that.
是指这个吗？比如，在协作方面，你喜欢那种合作的感觉
Be to it? Like, what in in terms of, like, you like the collaborative
如果一个问题太难解决，我喜欢先放一放继续前进
I like moving on from a problem if it's giving too much difficulty.
明白了
Got it.
但你需要那些有韧性和无所畏惧的人。我曾与这样的人合作过，当我想放弃时——因为我们尝试的第一种方法行不通，第二种方法也不行——他们却坚信不疑，并且准备好了第三、第四甚至第五种可行方案。最后我只能承认自己错了
But you need the people who have the tenacity and the the fearlessness. And I I've I've collaborated with with people like that where where I wanna give up, because the the first approach that we tried didn't work, the second one didn't approach. They're convinced, and they have the third, fourth, and the fifth of what works. And I'd have to eat my words. Okay.
我原以为这行不通，但确实，你一直都是对的
I didn't think this was gonna work, but, yes, you were right all along.
我们应该说明一下，对于不了解的人而言，你不仅以工作的卓越性闻名，还以惊人的生产力著称——发表了大量高质量论文。这说明能够在不同课题间灵活切换确实有其优势
And we should say for people who 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.
是的。这对我很有效。当然，也有人非常高效且专注于深度研究。我认为每个人都必须找到适合自己的工作方式
Yeah. It works for me. Yeah. I mean, but there are also people who are very productive, and they they focus very deeply on yeah. I think everyone has to find their own workflow.
数学中一个令人遗憾的地方在于，我们的数学教学采用了一种'一刀切'的方式。我们有固定的课程体系等等。我的意思是，也许如果你参加数学竞赛之类的活动，会获得略有不同的体验。但我认为很多人直到很晚——通常是为时已晚——才找到自己天生的数学语言。所以他们放弃了数学，并且因为老师试图用他们不喜欢的一种数学教学方式而留下了糟糕的体验。
Like, one thing which is a shame in mathematics is that we have mathematics is sort of a one size fits all approach to teaching mathematics. 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, get a slightly different experience. But I think many people, they don't find their their native math language 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 them one way to do mathematics that they they don't like it.
我的理论是，人类并非天生就拥有专门的数学大脑中枢。进化赋予了我们视觉中枢、语言中枢和其他一些经过打磨的中枢，但我们并没有与生俱来的数学感知能力。不过，我们其他的大脑中枢足够复杂，不同的人可以重新利用大脑的其他区域来进行数学思考。所以有些人学会了如何运用视觉中枢做数学，他们在数学思考时非常视觉化；有些人则重新利用了语言中枢，他们思考时非常符号化。
My theory is that humans don't come evolution has not given us a math center of a brain directly. We have a vision center and a language center and some other centers which have evolutions honed, but we it doesn't we don't have innate sense of mathematics. But our other centers are sophisticated enough that different people we we we can repurpose other areas of our brain to do mathematics. So some people have figured out how to use the visual center to do mathematics, and so they think think very visually when they do mathematics. Some people have repurposed their their language center, and they think very symbolically.
你知道，有些人如果非常竞争性强且喜欢游戏，大脑中有一个部分非常擅长解决谜题和游戏，这个部分可以被重新利用。但是，当我与其他数学家交流时，我能察觉到他们使用的思维方式与我的有些不同。我的意思是，并非完全不同，但他们可能更偏好视觉化。比如我自己就不太偏好视觉化，我个人需要很多视觉辅助工具。
You know, some people like if if they are very competitive and they they like gaming, there's a type there's this part of your brain that's very good at at at at solving puzzles and games, and and and that can be repurposed. But, like, when I talk to other mathematicians, you know, they don't quite think that I can tell that they're using somewhat different styles of of thinking than I am. I mean, not not disjoint, but they they may prefer visual. Like, I'm I I I don't like to prefer visual so much. I I need lots of visual aids myself.
你知道，数学提供了一种共同语言，但即使我们以不同的方式思考，仍然可以相互交流。但是你
You know, mathematics provides a common language, but we can still talk to each other even if we are thinking in in different ways. But you
能察觉到在思考过程中使用了不同的子系统。
could tell there's a different set of subsystems being used in the thinking process.
就像，他们走了不同的路径。他们在我挣扎的地方非常快，反之亦然，但最终都达到了相同的目标。是的，这很美妙。但是，除非你有私人导师之类的，否则我们的教育方式——
Like, they they take different paths. They're very quick at things that I struggle with and vice versa, and yet they still get to the same goal. Yeah. That's beautiful. And, yeah, but, I mean, the way we educate unless you have, like, a personalized tutor or something.
我的意思是，教育在某种程度上由于财务规模必须大规模生产。你知道，你必须教30个孩子。他们有30种不同的风格。你不可能用30种不同的方式来教学。
I mean, education sort of just financial scale has to be mass produced. You know, you have to teach the 30 kids. You know, they have 30 different styles. You can't you can't teach 30 different ways.
关于这个话题，对于那些在数学上遇到困难但感兴趣并希望提高的年轻学生，你会给出什么建议？在这个复杂的教育背景下有什么建议吗？你会怎么说？
On that topic, what advice would you give to students, 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. Complicated educational context? What what would you Yeah.
这是个棘手的问题。一个好处是现在课堂外有很多数学拓展的资源。嗯。在我那个时代，已经有数学竞赛了。你知道，图书馆里也有流行的数学书籍。
It's a tricky problem. One nice thing is that it there are now lots of sources for my faculty enrichment outside the classroom. Mhmm. So in in in my day, there already, are math competitions. You know, they're also, like, popular math books in the library.
嗯。你知道吗？但现在有了YouTube。有专门讨论数学谜题的论坛。数学也出现在其他地方，比如有些爱好者为了娱乐玩扑克。
Mhmm. You know? But but now you have, you know, YouTube. You there there are forums just devoted to solving, you know, math puzzles. And and math shows up in in other places, you know, like, for example, there there are hobbyists who play poker for fun.
他们出于非常具体的原因，对特定的概率问题感兴趣。是的。实际上在扑克、国际象棋、棒球等领域都有业余概率学家的社群。我是说，数学无处不在。
And they they, you know, they, for very specific reasons, are are interested in very specific probability questions. Yes. And and they they actually this community of amateur probabilists in in in in poker, in chess, in baseball. I mean, there's there's there's yeah. There's math all over the place.
我实际上希望借助Lean等新工具，我们可以将更广泛的公众纳入数学研究项目。这目前几乎完全没有发生。在科学领域，公民科学有一定空间。比如天文学家有业余爱好者发现彗星，生物学家有人能识别蝴蝶等等。在数学领域，也有少量活动让业余数学家能够发现新素数等等。
And I'm I'm I'm hoping actually with the with these new sort of tools of for lean and so forth that actually we can incorporate the broader public into math research projects. Like, this is almost is is doesn't happen at all currently. So in the sciences, there is some scope for citizen science. Like, astronomers, they're amateurs who would discover comets, and there's biologists, the people who could identify butterflies and so forth. And in method in there are a small number of activities where amateur mathematicians can, like, discover new primes and so forth.
但以前因为需要验证每一个贡献，大多数数学研究项目无法从公众输入中获益。事实上，这只会耗时，因为需要错误检查等等。但你知道，这些形式化项目的一个特点是它们正在吸引更多人参与。我相信高中生已经为一些形式化项目做出了贡献，比如为Dimethylib做出了贡献。你知道，你不需要拥有博士学位就能处理一个原子级别的问题。
But but previously, because we had to verify every single contribution, 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. But, you know, one thing about these formalization projects is that they are bringing together more bringing in more people. So I'm sure the high school students have already contributed to some of these formalizing projects who contributed to dimethylib. You know, you don't need to be a PhD holder to just work on one atomic thing.
这种形式化在第一步也向编程社区开放了。是的。那些已经熟悉编程的人。是的。编程似乎感觉上比数学更容易被大众接受。
There's something about the formalization here that also at at at the as a very first step opens it up to the programming community too. Yes. The people who are already comfortable Yes. With program. It seems like programming is somehow maybe just a feeling, but it feels more accessible to folks than math.
数学被视为一个，嗯，特别现代的数学，被视为一个极难进入的领域，而编程则不是，所以这可能只是一个入门点。
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.
你可以执行代码并获得结果。你知道，你可以很快地打印出世界。是的。你知道，如果编程被教成一个几乎完全理论性的学科，你只学习计算机科学、函数理论、例程等等。除了某些非常专业化的作业之外，你知道，比如你在周末为了乐趣而编程。
You can execute code and you can get results. You know, you can print out the world pretty quickly. Yeah. You know, like, if if programming was taught as an almost entirely theoretical subject, where you're just taught the the computer science, the the theory of functions and and and and and routines and so forth. And and outside of some some very specialized homework assignments, you know, like, you program, like, on the weekend for fun.
是的。或者对。那会被认为和数学一样难。
Yeah. Or yeah. That would be as considered as hard as math.
嗯。
Mhmm.
是的。所以正如我所说，你知道，有一些非数学家的社区，他们为了某些非常具体的目的而运用数学，比如优化他们的扑克游戏。对他们来说，数学就变得有趣了。
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 their poker game. And and for them, then math becomes fun for them.
你会给年轻人什么一般性的建议，关于如何选择职业，如何找到自我，比如他们可能
What advice would you give in general to young people how to pick a career, how to find themselves, like, they could be
擅长什么？这是一个非常非常难的问题。是的。现在世界上有很多不确定性。你知道吗？
good at? It's a tough tough tough question. Yeah. So there's a lot of uncertainty now in the world. You know?
我的意思是，战后有一段时期，至少在西方，如果你来自一个好的社会阶层，你知道，有一条非常稳定的通往好职业的道路。你上大学，接受教育，选择一个专业，然后坚持下去。但这正变得越来越成为过去式。
I mean, I I there was this period after the war where, least in the West, you know, if you came from a good demographic, you, you know, like, you there was a very stable path to a good career. You go to college. You get an education. You pick one profession, and and you stick to it. Just becoming much more a thing of the past.
所以我认为你必须具备适应性和灵活性。人们需要掌握可迁移的技能。比如学习一门特定的编程语言或某个特定的数学领域本身并不是特别可迁移的技能，但懂得如何用抽象概念推理或如何在出问题时解决问题才是。总之，我认为这些能力我们仍然需要。
So I think you just have to be adaptable and flexible. I think people have to get skills that are transferable. You know, like like learning one specific programming language or one specific subject of mathematics or something. It's it's it's that itself is not a super transferable skill, but sort of knowing how to reason with with abstract concepts or how to problem solve when things go wrong. So anyway, these are things which I think we will still need.
即使我们的工具变得越来越好，你知道，你将会与AI工具等一起工作。
Even as our tools get get better, you know, you'll you'll be working with AI sports and so forth.
但实际上，你是一个有趣的案例研究。我的意思是，你就像是当今最伟大的数学家之一，对吧？你有一套做事的方式，然后突然开始学习——首先，你一直在学习新领域。是的，但你学习了Lean。
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, start learning I mean, first of all, you kept learning new fields Yeah. But you learned lean.
那可不是一件小事。对很多人来说，那是一个极其不舒服的跨越，对吧？
That's not that's a nontrivial thing to learn. Like, that's a Yeah. That's a for for a lot of people, that's an extremely uncomfortable leap to take. Right?
是的。很多数学家...首先，我一直对做数学的新方法感兴趣。我觉得我们现在做事的很多方式效率低下。我和我的同事们花很多时间做非常常规的计算，或者做其他数学家立刻知道怎么做而我们不知道的事情。
Yeah. A lot of mathematicians. First of I've I've always been interested in new ways to do mathematics. I I I feel like a lot of the ways we do things right now are inefficient. I I I spend me and my colleagues, we 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 them.
为什么我们不能搜索并快速得到回应等等？这就是为什么我一直对探索新的工作流程感兴趣。大约四五年前，我在一个委员会，我们需要为数学研究所征集有趣研讨会的主意。当时，Peter Scholze刚刚形式化了他的一个新定理，计算机辅助证明领域还有其他一些看起来相当有趣的发展。我说，哦，我们应该办一个关于这个的研讨会。
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 Schulzer had just formalized one of his 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 should run a workshop on this.
这是个不错的主意。当时我对这个想法有点过于热衷，结果就被指派去实际负责这个项目了。于是我和其他一些人一起，包括凯文、乔丹·艾伦伯格等，共同完成了这个项目。这是一个相当成功的尝试。我们聚集了一批数学家、计算机科学家和其他领域的专家，共同了解了该领域的最新进展。
This would be a good idea. And then I was a bit too enthusiastic about this idea, so I I got voluntold to actually run it. So I did with a bunch of other people, Kevin and Jordan Ellenberg and and a bunch of other people. And it was it was a nice success. We brought together a bunch of mathematicians and computer scientists and other people, and and we got up to speed on state of the art.
其中有很多非常有趣的发展，大多数数学家都不知道正在发生什么，有很多很好的概念验证。你知道，这些都只是未来发展的预兆。这是在ChatGPT出现之前，但当时甚至已经有关于语言模型及其未来潜力的讨论。这让我对这个主题感到兴奋，于是我开始做讲座，告诉大家既然我组织了这次会议，现在应该有更多人开始关注这个领域。
And it was really interesting developments that that most mathematicians didn't know what was going on, that lots of nice proofs of concept. You know, it's just sort of hints of of what was going to happen. This is just before chat GBT, but there was even then there was one talk about language models and the potential capability of of those in the future. So that got me excited about the subject. So I started giving talks about this is something which more of us should start looking at now that I arranged the conference.
然后ChatGPT问世了，突然间人工智能无处不在。于是我接受了很多关于这个话题的采访，特别是关于人工智能与形式化证明辅助之间的互动。我说，是的，它们应该结合起来。这简直是完美的协同效应。
And then chat GPT came out, and, like, suddenly AI was everywhere. And so I got interviewed a lot about about this topic, and in particular, the interaction between AI and and formal proof of assistance. I said, yeah. They they should be combined. This this is this is this is perfect synergy to to happen here.
在某个时刻，我意识到不能光说不练，必须付诸实践。你知道，我不从事机器学习工作，也不从事形式化验证工作，我能依靠权威说'我是资深数学家，相信我，这将会改变证明方式'的程度是有限的，尤其当我自己并不亲自实践的时候。
And at some point, 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 and I don't work in truth formalization, and there's a limit to how much I can just rely on authority and say, you know, I I I'm a I'm a warm mathematician. Just trust me, you know, when I say that this is gonna change my phonics, and I'm not doing it any when I don't do it anyway myself.
所以我觉得我必须真正证明这一点。实际上，我投入的很多项目，一开始并不清楚会花费多少时间。往往是在项目进行到一半时我才意识到这一点，而那时我已经无法回头了。
So I thought I I had to actually justify it. Yeah. Lot of what I get into, actually, 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 waist deep in in in in a project that I I realized. By that point, I'm committed.
你愿意投身其中真是令人钦佩。在某种程度上成为一个初学者，对吧？或者面对一些初学者会遇到的挑战，对吗？
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?
是的。新概念、新的思维方式，还有，你知道，在某些方面不如别人。我想在那次谈话中，你可能是菲尔兹奖获奖数学家，但某个本科生在某些方面懂得比你多。
Yeah. New concepts, new ways of thinking, also, you know, sucking at a thing that others, I think I think in that talk, you could be a Fields medal winning mathematician, and an undergrad knows something better.
是的。我认为数学本质上——我的意思是，数学如今如此庞大，没有人能掌握所有现代数学。不可避免地，我们会犯错。而且，你知道，你不能仅仅靠虚张声势来掩盖错误，因为人们会要求看你的证明。如果你没有证明，那就是没有证明。
Yeah. 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, you know, you can't cover up your mistakes with just sort of bravado and and I mean, because people will ask for your proofs. And if you don't have the proofs, you don't have the proofs.
我不喜欢数学。
I don't love math.
是的。所以它确实让我们保持诚实。我的意思是，这并不是完美的万能药，但我认为我们确实比其他人更有承认错误的文化，因为我们总是被迫这样做。
Yeah. So it does keep us honest. I mean, not not I mean, you can still it's not a perfect panacea, but I think we do have more of a culture of admitting error than because we're forced to all the time.
一个荒谬的大问题。我再次为此道歉。谁是有史以来最伟大的数学家？也许是一位已经离世的。候选人有哪些？
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?
欧拉、高斯、牛顿、拉马努金、希尔伯特。
Euler, Gauss, Newton, Ramanujan, Hilbert.
首先，正如之前提到的，这存在一定的时间依赖性
So first of all, as as as mentioned before, like, there's there's some time dependence
但就目前而言。
But on the day.
是的。就像，如果你如果你如果你随着时间的推移累积地取得成果，比如说欧几里得，就像是，某种程度上算是吧。他是数学界的竞争者之一。然后在那之前可能有一些未具名的匿名信息，你知道的，就是那些提出数字概念的人。
Yeah. Like like, if if you if you if you pop cumulatively over time, for example, Euclid, like like, sort of like Yeah. Is is is one of the league contenders. And then maybe some unnamed anonymous messages before that, you know, whoever came up with the concept of of numbers. You know?
你明白吗？
You know?
今天的数学家们还能感受到希尔伯特的影响吗？哦，当然。直接体现在二十世纪发生的所有事情上？
Do mathematicians today still feel the impact of Hilbert? Just Oh, yeah. Directly of what everything that's happened in the twentieth century?
是的。我们有希尔伯特空间。当然有很多以他命名的事物。数学的整体架构以及某些概念的引入。我的意思是，那23个问题极具影响力。
Yeah. I've got helper 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.
宣布哪些问题是难以解决的，这种声明本身就有种奇特的力量。
There's some strange power to the declaring which problems. Yeah. Are hard to solve, the statement of the open problems.
是的。我是说，你知道，这就是无处不在的旁观者效应。如果没有人说你应该做某件事，大家就会原地打转等待别人去行动，结果什么事都做不成。所以，实际上你必须教给数学专业本科生的一件事就是：你应该总是尝试做点什么。你会看到很多本科生在尝试数学问题时陷入瘫痪状态。
Yeah. I mean, you know, this is bystander effect in everywhere. Like, if if no one says you should do x, everyone just will mill around waiting for somebody else to to, to do something, and and, like, nothing gets done. So and and, like, it it's the one one thing that actually you have to teach undergraduates in mathematics is that you should always try something. So you see a lot of paralysis in undergraduate trying a math problem.
如果他们认识到有某种技术可以应用，他们会尝试使用。但有些问题他们看不到任何标准技术明显适用。常见的反应就是完全瘫痪：我不知道该做什么。或者我记得《辛普森一家》里有句名言。
If they recognize that there's a certain technique that 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. I or I think there's a quote from the Simpsons.
我已经尝试了所有方法，但依然毫无头绪。所以，接下来就是不管多愚蠢的方法都要尝试，事实上，方法越愚蠢越好。因为我觉得这几乎注定会失败，但失败的方式会很有启发性。比如，失败是因为你完全没有考虑到这个假设。哦，这个假设肯定是有用的。
I've tried nothing, and I'm all out of ideas. So, you know, like, the next step then is to try anything, like, no matter how stupid. And in fact, almost the stupid of the better, which, you know, I won't I think it could just almost guarantee to fail, but the way it fails is gonna be instructive. Like, it it fails because you you you're not at all taking to account this hypothesis. Oh, this hypothesis must be useful.
这是个线索。
That's a clue.
我记得你还在某个地方提到过这个迷人的方法，
I I think you also suggested somewhere this this fascinating approach, which
这个方法让我印象深刻。我开始使用它，
really stuck with me. I started using it, and
确实很有效。我记得你说这叫结构化拖延。不对。对的。就是当你真的不想做某件事时，你就想象一件更不想做的事
it really works. I think you said it's called structured procrastination. No. Yes. It's when you really don't wanna do a thing, then you imagine a thing you don't wanna do more Yes.
没错。因为那件事比这个更糟糕。这样你就通过不去做更糟糕的事来拖延。对。对。
Yes. Because that's worse than that. And then in that way, you procrastinate by not doing the thing that's worse. Yeah. Yeah.
这是个很巧妙的小技巧。实际上真的很管用。
That's a nice it's a nice hack. It actually works.
是的。是的。我的意思是，任何事情都是这样，比如，你知道，心理学真的很重要。就像你和马拉松运动员等人交谈时，他们会讨论什么是最重要的。是他们的训练计划还是饮食等等？
Yeah. Yeah. There's I mean, with anything, like, you know, I mean, like, if psychology is really important. Like, you you you talk to athletes like marathon runners and so forth, you know, and they talk about what's the most important thing. Is it their training regimen or their diet and so forth?
实际上，很大程度上是心理学。你知道吗？就是欺骗自己认为形式是可行的，这样你才有动力去做。
Actually, so much of it is psychology. You know? Just tricking yourself to to think that the form is feasible so that you can be motivated to do it.
是否存在人类心智永远无法理解的事物？
Is there something our human mind will never be able to comprehend?
嗯，我猜某种程度上，作为一个数学家，我的意思是，这是一种推理。而且肯定存在一些你无法理解的大数。这是我首先想到的。
Well, I sort of I guess, a mathematician I mean, you know, like, it's a it's a deduction. And and it's very there must be some large number that you can't understand. Was the first thing that came to mind.
那么，但更广泛地说，是否存在我们心智的某种局限，即使有数学的帮助也无法突破？
So that, but even broadly, is there are we limp is there something about our mind that we're going to be limited even with the help of mathematics?
好吧。我的意思是，这取决于你愿意接受多少增强？比如，举个例子，如果我连纸笔都没有，没有任何技术手段。好吧，我不被允许使用黑板、笔和纸。
Well, okay. I mean, it's like, how much augmentation are you willing? Like like, for example, if if I didn't even have pen and paper, like, if I had no technology whatsoever. Okay. So I'm not allowed blackboard, pen, and paper.
没错。那样你已经比正常情况下受限多了。
Right. You're already much more limited than you would be.
极其有限。甚至语言，英语本身就是一种技术。这是一种已经被高度内化的技术。
Incredibly limited. Even language, the English language is a technology. It's it's one that's been very internalized.
所以你说得对。问题的表述方式确实不正确，因为早已不存在纯粹的独立个体人类。我们早已通过极其复杂精密的方式得到了增强，对吧？
So you're right. They're really the the the formulation of the problem is incorrect because there really is no longer a just a solo human. We're already augmented in extremely complicated, intricate ways. Right?
是的。没错。
Yeah. Yeah.
是的。所以我们已经是某种集体智慧了。
Yeah. So we're already like a collective intelligence.
对。是的。我想是这样。所以从原则上说，人类整体在其状态良好时拥有的智慧远超所有个体人类智慧的总和。当然也可能表现得更差。
Yes. Yeah. I guess. So humanity, plural, has much more intelligence in principle on his good days than than than the individual humans put together. It can all have less.
好的。不过确实如此。数学界这个集体，是一个极其超级智能的实体，没有任何单个数学家能够接近复制这种智能。你在一些问答分析网站上就能看到一点迹象，比如Math Overflow，那是数学版的Stack Overflow。
Okay. But yeah. So, yeah, math math the mathematical community, plural, is is is incredibly super intelligent entity that no single human mathematician can can come close to to to replicating. You see it a little bit on these, like, question analysis sites. So this math overflow, which is the math version of Stack Overflow.
嗯。有时候你会看到社区对非常困难的问题给出非常快速的回应。作为一个专家，观察这个过程其实是一种享受。
Mhmm. And, like, sometimes you get, like, this very quick responses to very difficult questions from the community. And it is it's it's a pleasure to watch, actually, as an as an expert.
我是那个网站的粉丝观众，只是欣赏那里不同人的才华。人们拥有的知识深度，以及愿意在特定问题上投入严谨和细致讨论的态度。看着这一切真的很酷，很有趣，几乎就像观看娱乐节目一样享受。
I'm a fan spectator of that of that site, just seeing the brilliance of the different people there. The depth of knowledge that people have, and the the willingness to engage in the in the rigor and the nuance over the particular question. It's pretty cool to watch. It's fun. It's almost like just fun to watch.
关于我们人类文明正在进行的这一切，是什么给了你希望？
What gives you hope about this whole thing we have going on, human civilization?
我认为，是的，年轻一代总是非常有创造力、热情且富有创新精神。与年轻学生一起工作是一种乐趣。科学的进步告诉我们，过去非常困难的问题可能会变得，你知道，变得像解决小问题一样简单。就像导航技术一样。
I think, yeah, the younger generation is always, like like, really creative and enthusiastic and and inventive. It's a pleasure working with with with young students. You know, the 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? And, I mean, like, it it was like navigation.
你知道吗？仅仅知道自己在星球上的位置曾经是个可怕的问题。人们因此丧命，或者损失财富，因为他们无法导航。而现在我们口袋里的设备能自动为我们解决这个问题，就像这已经完全不是问题了。
You know? Just just knowing where you were on on the planet was this horrendous problem. People people died, you know, or or lost fortunes because they couldn't navigate. You know? And we have devices in our pockets that do this automatically for us, like, as a completely solved problem.
对吧？所以现在对我们来说似乎不可行的事情，未来可能只是简单的课后练习。嗯。
You know? So things that seem unfeasible for us now could be maybe just homework exercises for me. Mhmm.
是的。我觉得生命有限性最令人悲伤的一点是，我将无法看到我们文明创造的所有酷炫事物，因为想想未来100年、两百年后的景象，只是想象一下两百年后的世界会是什么样子。
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 in in the next 100, two hundred years, just imagine showing showing up in two hundred years.
是的。不过已经发生了很多了不起的事情。你知道吗？比如，如果你能回到过去和你少年时的自己交谈，我的意思是... 光是互联网和我们的AI技术就足够震撼了。
Yeah. Well, already plenty has happened. You know? Like, if if you could go back in time and and talk to your your teenage self or something, you know, I mean Yeah. Just the Internet and and and our AI.
我的意思是，我理解他们已经开始内化并认为，是的。当然，人工智能可以理解我们的声音，并对任何问题给出合理但稍微不准确的答案。是的，即使在两年前，这已经令人震惊了。
I mean, I I get they've they've been into they're beginning to internalize and says, yeah. Of course, an AI can understand our voice and and give reasonable, you know, slightly incorrect answers to to any question. Yep, I this was mind blowing even two years ago.
而在当下，在互联网上观看这些戏剧性场面非常有趣。人们很快就对一切习以为常，然后我们人类似乎总是通过戏剧来自娱自乐。无论创造出什么，总有人要持一种观点，另一个人则持相反观点，彼此争论。但当你纵观全局，我的意思是，即使在机器人技术的进步中也是如此。是的，退一步看，真的会让人感叹，哇。
And in the moment, it's hilarious to watch on the Internet and so on, the the drama. People take everything for granted very quickly, and then they we humans seem to entertain ourselves with drama. 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, just even in progress of robotics Yeah. Just to take a step back and be like, wow.
人类能够创造出这一切的方式真的很美。
This is beautiful the way humans are able to create this.
是的。当基础设施和文化健康时，你知道，人类社区可以比其中的个体更加智能、成熟和理性。
Yeah. When the infrastructure and and the culture is is healthy, you know, the community of humans can be so much more intelligent and mature and and and rational than the individuals within it.
嗯，有一个地方我总是能指望找到理性，那就是你博客的评论区，我是你的粉丝。那里有很多非常聪明的人，当然也要感谢你在博客上分享这些想法。我无法告诉你，你今天能花时间与我交流，我有多么荣幸。我期待这次对话很久了。Terry，我是你的超级粉丝。
Well, one place I can always count on rationality is the comment 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 honored I am that you would spend your time with me today. I was looking forward to this for a long time. Terry, I'm a huge fan.
你激励了我。你激励了数百万人。非常感谢你的谈话。哦，谢谢。这是我的荣幸。
You inspire me. You inspire millions of people. Thank you so much for talking. Oh, thank you. Was a pleasure.
感谢收听与特伦斯·陶的这次对话。要支持本播客，请查看描述中的赞助商或访问 lexfreedman.com/sponsors。现在，让我用伽利略·伽利莱的一句话作为结束：数学是上帝书写宇宙的语言。感谢收听，期待下次再见。
Thanks for listening to this conversation with Terrence Tau. To support this podcast, please check out our sponsors in the description or at lexfreedman.com/sponsors. And now, let me leave you with some words from Galileo Galilei. Mathematics is a language with which God has written the universe. Thank you for listening and hope to see you next time.
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