AI 与数学的未来：对话 Grant Sanderson

自从那个难以捉摸的纳维-斯托克斯问题可能被解决以来——这里加个星号——似乎每个人都在谈论 AI 和数学。我的意思是，他们到处都在谈论 AI，对吧？但它在数学界确实让人感受到了它的存在。我最近和几个人聊过这个话题，其中一位是 Grant Sanderson。Grant 运营着极受欢迎且无疑非常优秀的 YouTube 频道 3Blue1Brown。我相信你们大多数人都看过。但 Grant 也是一个聪明、深思熟虑的人，他花大量时间思考数学的各个方面，无论是证明、传播还是教育。他还刚写了一篇精彩的文章，关于数学家的未来可能是什么样子。这篇文章客座发布在陶哲轩的博客上，当然，我会在下面附上链接。但在我开始问 Grant 那些宏观问题之前，我想了解 AI 是否对他个人产生了影响，作为一个制作 YouTube 视频的人。

Since the possible solving of the elusive Navier-Stokes problem—insert asterisk here—it seems everyone's talking about AI and mathematics. I mean, they're talking about AI everywhere, aren't they? But it's certainly making its presence felt in the world of math. I've been talking to a few people about it recently, and one of them is Grant Sanderson. Grant runs the wildly popular and undoubtedly excellent YouTube channel 3Blue1Brown. I'm sure most of you watch it. But Grant's also just a clever, thoughtful guy who spends a lot of time thinking about all aspects of math, be it proofs, communication, education. He's also just written an excellent article about what the future might look like for mathematicians. It was guest-posted on Terry Tao's blog, and of course, I will link to that below. But before I started asking Grant about all that big-picture stuff, I wanted to find out whether AI was having an impact on him personally, as a guy who makes YouTube videos.

作为一个数学传播者，你知道，一个靠制作数学视频和传播数学为生的人，AI 对你有什么影响？你在使用它吗？它是否在竞争等方面给你带来了问题？比如，告诉我 AI 如何影响 Grant Sanderson 和 3Blue1Brown。

As a math communicator and like, you know, someone who makes your living from making videos about maths and communicating maths, how has AI affected you? Are you using it? Is it creating problems for you with competition and stuff like that? Like, tell me about how AI affects Grant Sanderson and 3Blue1Brown.

天哪，好吧。我觉得每次你以某种方式问 AI 时，都值得区分你具体指的是什么，因为它可能意味着 20 种不同的事情。所以，在这方面，比如将其作为创作流程的一部分，是这个问题相关的一种方式。一种是作为视频的主题，因为我有一个关于神经网络的系列，对类似内容有需求。还有，嗯，我们不会称他们为 3Blue1Brown 的模仿者，但我们会说，有些人想要制作一个端到端的 LLM 在 YouTube 上创建数学视频，自然的方式是使用我用于视频的工具，因为它是程序化的，而这是 LLM 非常擅长的。然后，就像你在录制前描述的那样，它引起了如此大的共鸣，你说你感到某种义务去评论 AI 在数学中的进展，而你说“我有点想继续做我平常的事情”。所以，在我脑海中，它如何影响我的最显著的事情之一是，我有一个项目已经进行了大约一年半，收集数学家的采访片段，了解他们对 AI 和数学的看法，意图制作一部纪录片类型的东西，我会制作的。这感觉像是一种负担，一种我不得不做的事情，就像“好吧，好吧，我必须利用这个。我必须在这个领域发表一些评论。”这与我所热爱的——展示数学的美和深入问题——有点相悖。我想你问题的精神可能在于第一类，即在制作中使用它。即使这样，也分为——原谅我把它分解成这棵无限的树——但它分为许多不同的事情，因为你有 AI 工具来翻译你的内容。你有图像生成和你知道的视频生成之类的东西。你有写脚本。我的意思是，你有为它创建代码，为它做研究。

Man, okay. I feel like every time that you ask about AI in some way, it's worth separating out what specifically you mean, cuz it can mean like 20 different things. And so, on this front, you've got like using it as part of the creation pipeline is like one way that question would be relevant. One is like as a topic matter for videos, cuz I have a series about neural networks, there's like demand for more things like that. There is um uh we won't call them 3Blue1Brown copycats, but we'll we'll say like people who want to make an end-to-end LLM creates a math video on YouTube, a natural way to do that is by using the tool that I use for the videos because it's programmatic and like that's something LLMs are very good at. And then there's like you actually were describing this before we recorded and it resonated so much where you said you feel a certain obligation to like comment about the goings-on of AI in math and you're like, "I kind of want to just keep doing my normal thing." And so one of the most salient things in my mind for how it's affected it is I've had this project going for like over a year and a half of collecting interview footage with mathematicians for their thoughts on AI and math with the intent of making like a documentary type thing with it, which I will make. It kind of feels like a like a burden on the shoulders of a thing that I kind of have to do where like, "Okay, okay, I've got a I've got to like make use of this. I've got to make some kind of comment in this space." Which feels slightly at odds with just what I love doing, showing the beauty of math and diving into a problem. I think the spirit of your question was maybe in that first category of like using it in production. And even that splits into like forgive me for just like breaking this all down into this infinite tree, but that splits down into so many different things cuz you've got there are tools for AI to like translate your content. You've got like image generation and you know, video generation type stuff. You get like writing the script. I mean you've got like creating the code for it, doing research for it.

是的。

Yeah.

所以其中一些似乎只是有害的，对吧？比如用它来生成视频或图像。第一，结果并不好。但即使它们可能还行，这只是一个第三轨，对吧？比如与之相关的品牌损害根本不值得。今年，我从去年独自工作转变为——今年的一个显著区别是我现在有两个艺术家和我一起工作。一个是实习生，一个更接近全职。我认为和他们一起工作比试图摆弄任何那些工具要有趣得多。结果更好，而且没有争议或其他问题。如果有的话，更像是竖起一面旗帜，说“不，我们确实喜欢手工制作，为新的艺术作品倾注爱心地创作。”

So some of these seem like some of them are just toxic, right? Like using it for video generation or image generation. One, the results just like aren't great. But even to the extent that they may be okay, it's just a third rail, right? Like the brand damage associated with that is like nowhere near worth it. This year so I went from working solo last year where like a notable difference this year is like I have two artists who like work with me right now. One is an intern, one is something closer to full-time. And it's just much more fun I think to to work with them than to try to like fiddle around with any of those tools. The result is just better and you don't get that there's no like controversy or anything around it. And if anything, it's kind of like nicer to plant a flag in the sand of like, "No, we do like hand-crafted, like lovingly created artwork for the new artwork that there is."

显然，你的视频本质非常，嗯，你知道，它们是用软件和计算机动画制作的。你是否发现人们还是指责你用 AI 制作它们，尽管你是在手工制作？

Obviously, the nature of your videos that are very, um, you know, they are made with software and computer animation. Are you finding people are accusing you of making them with AI anyway, despite the fact you're handcrafting them?

我没有，我没见过那种指责。我会说，好吧，这里有一件事变了。我觉得我的风格默认情况下，它看起来不那么人性化，或者有点不那么，嗯，它不那么本质上是人性的。在其历史的很多部分，这是有意为之的。我们处于纯粹的抽象和数学领域，图形就像所讨论数学的精确表示。这本身是令人愉悦的。那是你可以拥有的一种风格。我发现自己

I don't I haven't seen that flavor of accusation. I will say, okay, here's a thing that's changed. I feel like my style can just like by default, it just looks less human or it's a little bit less, um, it's less intrinsically human. And that was kind of deliberate for so much of its history. We were on a pure land of abstraction and math and the graphics are like a a exact representation of the mathematics being discussed. And that's pleasing in its own way. That's the kind of style that you can have. I have found myself

这几乎是你的独特卖点。

That's your USP, almost.

对，但现在我觉得我不得不加入一些东西，其实我不必。没有人指责它是 AI 生成的，但为了潜意识地向观众表明这是精心制作的，我会故意做一些事情，比如打破第四面墙或出现在镜头中。所以，例如，嗯，加入更多艺术作品。那是我本来就想做的事情，给出这些事物背后的一些人类故事，而不是仅仅用纯计算机图形展示数学，有一些手工制作的、艺术性的东西展示背后的人类。这部分是一种区分。另一件事，比如今年早些时候我做的一个视频，我希望开场镜头是幕后花絮，我在编码动画，我在引用等等，它起到了双重作用。第一，我在说“这是一个我非常兴奋的话题。”在制作这些动画时，有一种特定的感觉让我知道我会喜欢这个视频。另一个我想通过那个开场镜头达到的目的是那种潜意识的点头，就像你知道的，这是精心制作的，对吧？在 2019 年，我可能不会以那种方式思考。

Right, but like now I I I feel like I have to throw things in I don't have to. There's no one who's like accusing it of being AI generated, but for the sake of subconsciously making clear to an audience member that this was lovingly crafted, I will deliberately do things to like break the fourth wall or like be in there. So, for example, um, having more artwork. That's the thing I kind of wanted to do anyway, give some of the human stories behind these things rather than it being just pure like computer graphics showing the math, have some like, you know, hand-crafted, uh, artistic stuff showing the humans behind it. That partly is a little bit of a separation. Another thing, like there was this video I did earlier this year where I wanted the opening shot to kind of be a behind the scenes of me coding up the animations and me like referencing and and and it was just doing double duty. One, I was saying like, "This was a topic I was so excited about." Like there's a certain feeling while I'm like making these animations that makes me know I'm going to love the video. The other thing I wanted to accomplish with that opening shot was that little subconscious nod that like, you know, this was crafted, right? In a way that I wouldn't have maybe been thinking in those terms in 2019.

所以，视频本身是一个第三轨，你目前不愿意让 AI 触碰它。如果有的话，你愿意用 AI 做什么？

So, the video itself is a third-rail and you're not willing to you're not willing to let AI touch that at the moment. What are you willing to use AI for, if anything?

是的，所以我认为，嗯，在创建动画方面，我可以使用智能体式工具，比如，如果我给出一个动画的指令，我想用英语制作，让它写出 Manim 代码。

Yeah, so I think um on the in in creating animations, I would be fine using agentic tools to like um if I if I give the instruction on an animation that I want made um in English for it to like write the Manim code for that.

话虽如此，对我来说，直接手写代码其实更容易、也更快，因为逐行都有很多设计决策。对于这个工具 Manim 的基础设施工作，我会用它来帮忙。我觉得过去 12 个月里，关于智能体式 AI 到底是对你已经擅长的东西更有用，还是对你还不擅长的东西最有用，发生了一个真正的转变。

That said, it's actually easier or faster for me to just code it by hand, because there are a lot of design decisions line by line. For infrastructural work on this tool, Manim, I will use it for helping with that. I think in the last 12 months there was a real shift on whether agentic AI was useful for things that you're already an expert in versus being most useful for stuff that you're not yet an expert in.

举个例子，我有个拖了很久的后端重构一直想做。这比你也许想听的更琐碎，但 Apple 停止支持 OpenGL 了，那是个做计算机图形的框架，我就得改后端的一些东西。基于他们不再支持，这就成了一颗定时炸弹，不知道什么时候会崩。要正确地做那个重构，学会我需要的那些东西来把它设计对，我觉得得花我一个月。但配合一个智能体式工具来做，结果只花了一个周末，而且它可能做得比我还好。所以我就想：“好吧，这真不错。”

As an example, I had this long-standing back-end refactor that I wanted to do. This is more in the weeds than maybe you want, but Apple stopped supporting OpenGL, which is a framework for computer graphics, and I just had to change some things on the back end of it. There's this ticking time bomb on when is that going to break based on their lack of support. It would take me like a month, I think, to do that refactor properly, to learn the things that I need to design it right. But then doing it in conjunction with an agentic tool, it was instead like a weekend, and it probably did better than I would have. And so I'm like, "Okay, that was real nice."

但我在那件事里感受到的其实是，我和这个工具的关系并不是纯粹功利性的。不是我只想要一个能完成某个功能的工具。我从搞懂一个东西怎么运作中获得内在的快乐。我之所以写一个工具来做频道里的图形，而不是用别的东西，原因之一就是搞懂东西怎么运作本身带来的快乐。而这是我第一次真正感觉到，如果我为功利目的做优化，比如尽快拿到我想要的所有功能，我就不会理解它。我不得不刻意放慢速度，只为了确保我仍然理解整条流水线里发生了什么。你知道，也许这是一种过时的思维方式，也许这就像去琢磨 C 编译成的汇编代码，但这是我仍然想守住的东西。

But what I felt in that actually, my relationship with this tool is one where it's not strictly utilitarian. It's not just like I want a tool that does a function. I find internal pleasure in knowing how the thing works. One of the reasons I wrote a tool to do the graphics on the channel rather than using other stuff is just an internal joy from knowing how things work. And this was one of the first times I really felt that were I to optimize for utilitarian purpose, like get all the functionality that I want as fast as I want, I would not understand it. And I had to just deliberately go slower than I could have just to try to make sure that I still understand what's going on in that whole pipeline. Which, you know, maybe that's an antiquated way of thinking, maybe that's analogous to thinking about the assembly code that C compiles into, but it's a thing that I still want to hold on to.

另一个方面，研究在学新东西上很有意思，因为对一个你不太熟悉的想法先拿到一个概览非常有用。不过我觉得用这些工具最有用的方式之一，几乎是把它当成一种文献检索，让它把我指向人类写的、最好的那些描述。基本上就是把它当成一个更好用的 Google：一开始，好吧，也许有个新东西我想学，给我一个快速概览。但早期阶段，要问的是：关于这个，真正善于阐述的人精心写就的文档有哪些？

The other aspect, so research is an interesting one on learning new things, because it's very useful to get just an overview of an idea that you're not super familiar with. I think one of the most useful things to do with these tools though is use them almost as a kind of literature search and point me to the human-written stuff that's going to be the best description. Basically treat it as a better version of Google on first, okay, maybe there's a new thing I want to learn, give me a quick overview. But early on, it's like what are the deliberately crafted documents from actual expositors on this?

因为我觉得会发生的一件事是，如果你在学新东西，甚至不知道该问什么问题，而你问了一个稍微偏掉的问题，工具就会顺着你走，顺着你切入的角度跟你互动。而如果你从别人的框架开始，那个人是专家，已经想清楚该怎么解释，那么最能带来差别的学习步骤，是有人告诉你应该抱持的正确思维框架。也许它们某天会擅长这个，但目前更好的做法是问它我该读什么、我该看哪个视频，然后再跟模型追问一些澄清之类的问题。所以这是混合的，并不是说在这件事上你 100% 都待在聊天界面里。但没错，我确实会在学新东西时用它。

Because I think one of the things that'll happen is if you're learning something new and you don't even know the right questions to ask and you ask the slightly wrong question, the tool kind of runs with you and it kind of engages with you coming at it. Whereas if you just start with someone else's framing where that other person is an expert and they've thought through how to explain it, the relevant learning steps that make the most difference are the ones where someone is telling you the correct frame of mind to even have. And maybe they'll get good at that at some point, but at the moment it's just better to ask it what should I be reading, what video should I be watching, and then maybe have follow-up questions with the model for clarification and things like that. So it's mixed, it's not like you're 100% in the chat interface for that. But yeah, I do use it definitely on learning new things.

Grant，你非常成功，显然你的视频有那种观感，是因为你用来制作它们的软件。而你也非常开放，让别人使用它，结果就是很多其他频道和创作者做出看起来很像你的视频。这一定让人受宠若惊，但有时也令人沮丧。我不知道那是什么感觉。我想象那会挺奇怪的。但我能看出，能接触到这些软件和工具的人，可以用 AI 大量炮制 Three Blue One Brown 风格的视频。用体育的比喻来说，就是大举压上、淹没整个区域。

Grant, you've been so successful and obviously your videos have this look because of the software you create them with. And you've been really open about letting other people use that, and that's resulted in a lot of other channels and creators making videos that look a lot like yours. Which must be flattering, but also frustrating sometimes. I don't know what it feels like. I imagine it would feel strange. But I can see that people who have access to all this software and these tools could use AI to really churn out Three Blue One Brown-style videos. Really flood the zone, to use a sporting analogy.

是的，我想那会影响到你，因为不管它好不好，都会有大量看起来像你那种东西的内容。那么，这是让你担心的事吗？当所有这些别的东西涌进来时，找到你的内容、认出你的内容会变得更难。

Yeah, and that's going to impact you, I imagine, because even if it's good or not, there's just lots of stuff that looks like your kind of stuff. Like, is that something that worries you? It's going to be harder to find your content or recognize your content as all this other stuff just kind of deluges in.

我最大的担心不太在于竞争层面，也就是有更多数学内容在争夺人们的注意力。我觉得真正出来的那些东西并不太吸引人，人们也许会有点厌倦。更大的担心是它会拉低品牌。如果有些我做的内容就是不够好，那么人们看到我真正做的东西时的第一印象，就会和所有那些东西绑在一起，我觉得这我逃不掉。

My biggest worry is not too much on the competitive side of there being more total math content that is competing for people's attention. I think it's not super engaging, the stuff that does come out, and people get a little bit tired by that maybe. The bigger worry is just that it degrades the brand. If there's stuff that I make that's just not great, then the immediate impression that people will have in looking at the stuff that I actually make is tied to all of that stuff, which I don't think I can escape.

说实话，这在 AI 之前就是个问题了。我记得有个频道，天哪，叫 Math Maniac，发过一篇很有想法的帖子，讲 AI 生成的数学视频越来越充斥互联网。有些频道每天发一个视频，甚至每天发好几个。他们用我的工具，因为我觉得这又是让 LLM 生成视频最容易的方式之一，尤其是关于数学的，因为它全是代码。那个视频下面有条评论说：“说实话，Manim 垃圾已经是个问题有一阵子了。”他们的意思不一定是 AI 生成的，而只是人们在使用这个工具，但用的方式……就像我用它的时候，它不是……

And to be honest, this was a problem before AI. So I remember there was a thoughtful post by this channel, oh god, Math Maniac, about AI-generated math videos kind of populating the internet more and more. And there's channels that put out a video every day or multiple videos every day. They use my tool just because it's, I think again, one of the easiest ways to get an LLM to generate videos, especially about math, because it's all code. And there was a comment on that video that said, "To be honest, Manim slap has been a problem for a while." Which by which they meant not necessarily AI-generated, but just people using the tool, but in a way that... like when I use it, it's not...

但对还没把点连起来的人来说，Manim 是你创建的那个工具的名字。

But for people who haven't joined the dots yet, Manim is the name of the tool you created.

对，对。Manim 是这个工具的名字。

Yeah, yeah. Manim is the name of the tool.

对。对。

Yeah. Yeah.

如果你刚开始用它，最自然的做法，你可以让它写一个方程，然后有个特定的动画来呈现那个方程怎么写上去，或者做基本的图表之类的。而我会说，我看到人们使用它的方式里，90% 我觉得本来用 Keynote 就够了，或者用别的方式做大概会容易得多。我觉得它有用的地方，是我几乎在为某个特定视频给这个工具做扩展的时候。你知道，有个关于牛顿分形的主题。

If you're just getting started with it, the most natural things to do, you can have it maybe write an equation and there's a certain animation for how that equation gets written on, or do basic graphs and whatnot. And I would say 90% of the ways that I'll see people use it, I think it just should have been in Keynote, or it probably would have been a lot easier to do in some other way. Where I find it useful is if I'm almost adding to the tool for the specific video. You know, there's a topic about Newton's fractal.

而为了解释那个主题，你想在代码中实现底层的数学，然后让它在屏幕上反映出来。所以你是在用它构建一个东西，然后只是展示出来，而不是使用库中已有的东西来拼凑出简单的形状相互交互。所以甚至在 AI 之前，我可能就有这种——我不称之为恐惧，但脑子里有点小疙瘩——很多人用这个工具创作的东西，其实你用 PowerPoint 或 Keynote 就能做，也许感觉有点浪费他们的时间，但它也让互联网上充斥着很多看起来像我会做的视频，但不一定有那种内在精神，即让我们找到一块复杂的数学，只有通过用代码实现并看看屏幕上出现什么才能变得清晰。我可能没有很好地表达出它是什么，但确实有

And in order to even explain that topic, you want to implement the underlying math in the code and then let that be reflected with what's on screen. And so you're building a thing using it and then there's just something shown as opposed to using the pre-existing stuff from the library to piece together like simple shapes interacting with each other. Um so even before AI, I maybe had this I don't call it a fear, but a little niggling in my brain that a lot of people creating things using the tool that are just what you could have done with, you know, PowerPoint or Keynote, um maybe feels like a little bit of waste of their time, but it also it populates the internet with a lot of things that have the look of the videos that I'll make, but not necessarily with the underlying spirit of saying let's find a complicated piece of math that will only become clear by implementing it in code and seeing what shows up on the screen. And I'm not doing a good job articulating maybe what it is, but there's

那么让我问你这个问题，显然你脑子里有很多想法在翻腾，还有很多要处理。但作为一个内容创作者，作为一个 YouTube 人，你认为 AI 对你是净赢还是净输，是警钟，还是好时光？

Let me ask you this then, to sort of obviously there's a lot bouncing around your head and there's still a lot to process. But as a content creator, as a YouTube guy, are you seeing AI as like a net win for you or a net loss, an alarm bell, or is it like good times?

净赢。只要用对了，肯定是净赢。比如，我不会用它来生成音乐。我会用我的作曲家。我不会用它来做翻译。我更喜欢今年雇的人类翻译。不会用它来做 artwork。有很多事情，在某种意义上，我只是不用它来做那些整个公司存在就是为了用它的东西。但在那些有帮助的小地方，比如我们谈到的研究，或者让工具的基础设施工作更快，那是巨大的。我还认为有一堆小的后勤事务，比如审查合同或类似的事情，在那个规模上，你实际上不想去找律师，但你也不想花接下来的 30 分钟仔细看它。有上千种小方式让它变得更好，所以完全是净赢。我举一个具体的例子，关于一件我不会做的事情，它增加了视频中的细节关注，我觉得完全可以用 AI。我几天前做了一个关于国际数学奥林匹克问题的视频。我想让开场镜头描绘所有来自世界各地的学生飞往那一年举办地澳大利亚。如果我自己做，我可能会——我最终做的是——这是一种无关紧要的细节关注，没人会在意，但它让我开心，我喜欢它。我试图创建一个表格，基本上是关于每个学生最可能从哪里飞来，也许是相关国家的首都，或者如果他们有一个特定的训练营，那个城市在哪里，以及每个国家有多少学生。通常是六个，但不总是六个。那一年有哪些国家？所以，建立那个表格，这样我可以把它加载到动画中，使得在开场镜头中，从加拿大飞来的学生实际人数是正确的，而且可能是他们飞来的正确城市。无关紧要。如果我不能只是问一个工具来创建那个表格并完成所有这些，那就不值得我花时间。

Net win. Net win for sure as long as you use it right. Like so, I'm not going to use it to generate music. I'm going to use my composer. I'm not going to use it for translations. I prefer like the human translators that I've like hired in this year. Not going to use it for artwork. There's lots of things here where in some sense I'm just not using it for things that entire companies exist to use it for. But the little bits where it is helpful um on on research like we talked about or making infrastructural work on the tool faster, that's huge. Um I also think there's a just a bunch of tiny little logistical you know like getting a contract reviewed or things like this that it's like at that scale of you don't you don't actually want to go to a lawyer, but you don't want to like spend the next 30 minutes of your life looking too closely at it. There's just a thousand little ways that it makes things a little bit nicer such that to- totally net win. I'll give you one example concretely on a thing I wouldn't have done that added attention to detail in a video that's like use of AI I feel totally all right with. I I a couple days ago made a video about a problem from the International Math Olympiad. I wanted the opening shot to kind of give a depiction of all these students from around the world flying to Australia where it was hosted in that year. And if I was just making this on my own, I might have What was it What I ended up doing This is an attention to detail that does not matter, that nobody would care about, but it tickled me and I loved it. Um I tried to create a table basically of where specifically is it most likely each student would have been flying from, you know, maybe the capital of the relevant country or if they have a specific training camp, what's the city where that is, and how many students were coming from each country. It's typically six, but it's not always six. And what what were the countries that year? And so, building up that table such that I could load that into the animation such that in that opening shot, the actual number of students flying from, you know, Canada, uh, is correct and it's like probably the right city that they're flying from. Does not matter. Would not have been worth my time if I couldn't just ask, you know, a tool to like create that table and and do all of that.

这就像一个小彩蛋，你知道的，如果没有 AI，你不会费心去植入。是的。

It's like a little Easter egg that you know about that you wouldn't you wouldn't have bothered to implant if you didn't have AI. Yeah.

是的。这就是最大的胜利，那些如果像 2016 年那样有那么多摩擦你就不会做的事情，但它只是让事情变得更好一点。

Yeah. And that's that's the biggest wins here are the things that you wouldn't have done had it been as as had it had as much friction, um, as it would have in, you know, 2016, but it just makes things like a little bit nicer.

目前的情况如何？比如，你会如何向一个非数学家解释？就像你在咖啡店或酒吧的朋友，他们只是说：“哦天哪，我最近听说了很多关于 AI 和数学的事情，我听说了千年问题之类的。那么，AI 和数学到底怎么回事？你会如何解释我们现在所处的位置？”

What is the current state of play? Like, how would you explain it to a non-mathematician? Like, you know, a friend of yours at a coffee shop or at the bar and they're just saying, "Oh gosh, I've heard so much about AI and mathematics at the moment and I've heard about millennium problems and all sorts. Right, what the heck's going on with AI and mathematics? How would you explain where we're at now?"

做数学也许可以分解成多个不同的部分，但其中一些只是知道术语，知道结果。这就像事实的积累。另一方面是，找到洞察力来解决一些新问题。我认为你开始看到的是，这些模型正在解决一些相当严肃的问题，人们已经研究了一段时间。有时是与人类提示它们并基本上与之合作的结果。有时更多是人类没有真正输入很多数学知识，而只是试图鼓励它继续下去。所以今年我们开始看到对严肃问题的解决方案，人们认真研究过，如果没有这些模型可能不会发生。其中最引人注目的是纳维-斯托克斯方程相关的东西，伴随着争议。但今年早些时候还有很多非常有趣的结果，导致了这个。其他登上头条的包括所谓的单位距离猜想。这位名叫保罗·埃尔德什的人留下了很多开放问题，作为他众多合作的一部分。AI 公司开始把它当作一种基准，即这些模型能回答多少埃尔德什问题。由于几个原因，这不是一个好的基准，但确实有真正有趣的数学从中产生。我最喜欢的一个是这个。它在埃尔德什问题目录中的编号是 1196。它是关于所谓的原始集的。但是什么让它从当时其他结果中脱颖而出？它看起来不像反例，而其他一些 AI 结果看起来像是说数学家猜想 X 是真的，但事实证明 X 不是真的，这里有一个复杂的反例说明为什么。有时你从中学习。但最美味的数学是当某事是真的。你不太知道如何阐述为什么它是真的或形式化它，但然后有一个非常清晰的想法展示了它。对于埃尔德什问题编号 1196 提供的解决方案，我认为它真正地让周围数学部分的理解更清晰。而且有几位数学家喜欢它，它不只是以最漂亮干净的形式从模型中出来。

Doing math maybe can break down into multiple different parts, but some of it is simply knowing the the terms, knowing the results. It's like a uh uh a a factual accumulation. Another side is, um, finding an insight to solve some new problem. And I think what you are starting to see is, uh, solutions to problems that are quite serious that people had worked on for a while that are coming out of these models. Sometimes as a result of an engagement with a human prompting them and, uh, basically working with it. Sometimes in more of a way where the human doesn't really input a lot of mathematical knowledge. It's more just that like trying to encourage it to keep going. And so this year we've started to see solutions to serious problems that people had seriously worked on that probably would not have happened without the models. The most headline gaining of which was the the Navier-Stokes stuff surrounded in the controversy that it had. But there were there were lots of really interesting ones even earlier in this year kind of leading up to that. Others that had made headlines included something called the the unit distance conjecture. This particular man named Paul Erdős had a whole bunch of open questions that he left as part of his many many collaborations. And the AI companies kind of started treating it as a benchmark of sorts on you know how many of these Erdős problems can these models answer. Which is not a great benchmark for a couple of reasons, but there was there was legitimately interesting math that came out of that. One of my favorite ones was this problem. It had the number 1196 in this catalog of Erdős problems. And it is about these things called primitive sets. But what made made it stand out from other results around then? It didn't look like a counterexample where some of the other AI results looked like saying mathematicians conjectured that X was true, but it turns out X is not true and here's this complicated counterexample showing why. Sometimes you learn something from that. But the the most delicious math is when something is true. You don't quite know how to articulate why it's true or formalize it, but then there's a really clean idea that shows it. And the the solution provided for this problem number 1196 of Erdős, it was it was something that I think genuinely made for a cleaner understanding of the surrounding bits of math. And there were a few mathematicians that like it did not just come out of the model in the it's nicest clean form.

但谁认识到，“嘿，这个想法在其他方面也有用。”并围绕它组织了一篇论文，如果你问：“2025 年与 2026 年人类对所谓原始集的理解是什么？”到 2026 年底，这真的很棒。它非常清晰。有一个非常好的想法。我认为可以说这个想法来自 AI。现在，有些地方有点棘手，很多新结果乍一看像是来自模型，但如果你更深入地搜索，会发现早期文献中有未被引用的种子，或者很可能是模型基于别人的想法，只是加了一块小石头来完成建筑。

But who recognized, "Hey, this idea is useful in other ways." And had a paper assembled around it that if you were to ask, "What is humanity's understanding of these things called primitive sets in 2025 versus 2026?" At the other end of 2026, it's really nice. It's like really clean. There's this really nice idea. And I think it's fair to say that idea came from the AIs. Now, where there's some where this gets a little bit tricky is a lot of the new results look at first like they came out of the model, but then if you search more deeply, there's seeds of it in earlier literature that are not attributed or it's probably the case that the model was working from someone else's idea and was just adding like a little stone on top to complete the building.

数学不一直是这样吗，Grant？我知道署名很重要，但是的。

Hasn't mathematics always been that way, Grant? I know attribution is important, but yeah.

是的。是的，是的。但所以缺乏署名是这里的关键部分，对吧？所以就像是的，数学一直是这样，就像添加最后一块石头的人，但那个人总是知道之前的石头是什么，他们会指出那些石头是什么，然后人们可以看着它们并欣赏。变得更难的是，特别是如果它被烘焙到模型的权重中，而不是在上下文中做的事情，可能的情况是，它站在巨人的肩膀上，但不知道要署名哪个巨人。所以这里有一个具体的例子。其中一个头条新闻的结果是对所谓雅可比猜想的一个反例。嗯，这就是结果发布的方式。有些人真的很不喜欢这个作为推文，因为你真的可以把结果放在一条推文中，因为它基本上是一个多项式。因为猜想说任何具有如此这般性质的多项式将具有如此这般的其他性质。它只是展示了一个反例，只是一堆数学。看起来就像是模型凭空抽出来的。就像，“哇，你给了我这个问题。”人们真的在试图证明这样的反例不存在或找到一个没人找到的，但这个就像凭空抽出来的。实际上有一篇来自俄罗斯出版物的论文，大约在 1999 年，它没有那个多项式，但有一个低维变体，它不完全作为这个猜想的反例，因为它有一个 1/Y 项，像一个奇点。嗯，很明显，那条推文中 Claude Fable 发现的东西可能是那个的变体，他们解决了问题。实质性的结果但完全没有署名它所基于的东西。另一个例子是 OpenAI 在所有 Navier-Stokes 东西之前有一个帖子，你知道，关于数学中 10 个重要问题，10 年没有进展，我们的模型解决了。嗯，我采访的一个人对那个标题的最初框架非常不满，因为他说其中一个问题关于所谓非 Sofic 群的存在性。他说你在说什么过去 10 年没有进展？我有同事在推进一个非常有前途的项目，很明显 OpenAI 的结果是在添加那个项目，所以他们我认为他们改变了 OpenAI 网站上他们如何框架的措辞。

Yes. Yeah, yeah. But so the lack of attribution is the relevant part here, right? So like yes, mathematics has always been it's like the person adding the final stone, but always that person like knows what the previous stones are, and they like point to what those stones are, and people can then look at them and appreciate it. What gets a little bit harder is, especially if it's baked into the weights of the model and it's not something that it's doing in context, it might be the case that it, you know, it's standing on the shoulders of giants, but it doesn't know what giant to attribute. And so the a a concrete example here. One of the headline-making results was a counterexample to something called the Jacobian conjecture. Um this was the nature of how the result was put out. Some people really disliked this was as a tweet because you can literally fit the like the result in a tweet cuz it's basically a polynomial. Cuz the conjecture says any polynomial that has such and such properties um will have such and such other properties. And it just demonstrated a counterexample, and it's just pile of math. The way that that looks is as if the model kind of pulled it from out of the blue. Like, "Whoa, you gave me this problem." People had really trying to uh either prove that such counterexamples don't exist or find one no one had, but this like pulled it out of the blue. There was actually this like paper from a Russian publication in something like 1999 that it didn't have that polynomial but it had like a lower dimensional variant of it that didn't quite serve as a counter example to this conjecture cuz it had like a one over Y term like a singularity. Uh and it's it's pretty clear that the thing that was in that tweet that Claude Fable had um like found was probably a variation on that and they like worked out the issue. Substantive result but totally unattributed like what it was building on top of. Another example would be OpenAI before all the Navier-Stokes stuff had this uh post about you know 10 significant problems in math that hadn't seen any progress for 10 years that like our model has solved. Um and I and one of the people I talked to took great offense at that initial framing of their headline cuz he's like one of these problems about the existence of something called a non-Sofic group. He's like what are you talking about didn't make any progress in the last 10 years? Like I have colleagues who were like pushing forward a very promising program on that and it's very clear that the result from OpenAI is like adding to that program that they were doing and so that I I I think they changed the the phrasing on their um on the OpenAI website for how they they framed that.

Grant，你能解释为什么署名重要，除了像你知道的礼仪和礼貌之类的以及数学的做事方式？

Grant, can you explain why the attribution is important other than like you know propriety and good manners and stuff like that and the way math is done?

是的，在学术界那是货币，对吧？我的意思是，它直接关系到人们的职业生涯，嗯，你知道，例如你的论文有多少引用之类的事情将是论文质量的指标。嗯，我认为是的，所以署名一般来说，你知道，这是好政策，好礼貌等等，但我认为对于学者来说，他们不存在于市场中，对吧？不是像正在发生的事情是人们在购买定理的证明，他们在外面出售那些定理的证明，并找到合适的价格等等。相反，相关货币是署名，就像人们基于你的结果并引用你这样做。我认为因此，即使在世界其他地方，如果有人基于你的工作而没有指出你，会有点冒犯。在那种背景下，它如此直接地关系到人们的职业轨迹，它不仅是有礼貌，而且有实质影响。

Yeah, in academia that's that's currency, right? I mean like that is it it it it bears directly on people's careers um you know for example how many citations your papers have and things like that will be a metric of the quality of the paper. Um I think yeah, so it attribution in general you know it's it's it's good policy it's good manners and everything but I think for an academic uh they don't exist in a market, right? It's not like what's happening is that um people are buying proofs of theorems and they're out there you know like selling those proofs of theorems and like finding the right price for them and everything. Instead, it's something where the relevant currency is is attribution, like people building on your results and then citing you for for doing so. And it's I think as a as a result, even though in the rest of the world it would be, you know, a little bit offensive if somebody was building on your work without pointing to you. In that context, where it bears so directly on people's career trajectories, it like not only is it good manners, but it also has material influence.

所以，它仍然与真理和知识的进步无关，但我们仍然在谈论一种人类系统，人们想要获得工作、晋升、加薪和引用计数，我并不是说这些事情不重要，我只是说它们对数学不是根本的，就像你知道的。

So, it still has no bearing on like truth and the advancement of knowledge, but it's we're still talking about like a kind of a human system here where people want to get jobs and promotions and pay rises and citation counts, which which I'm not I'm not saying those things aren't important, but I'm just saying they aren't fundamental to mathematics, like, you know.

嗯，我认为它确实与真理有点关系，因为所以，这是人们讨论的一个大话题，强调如何解决这些问题几乎无关紧要，通常你关心的是为了解决它而创造的新想法，然后这些新想法有溢出效应，这意味着仅仅看到解决方案往往不如理解导致该解决方案的思维轨迹有价值。而那个思维轨迹的一个相关部分是你所基于的工作体系，重要的地方在于你说，比如我们为什么关心解决这个 Navier-Stokes 问题或找到非 Sofic 群或雅可比猜想，无论这些东西是什么。几乎从来不是你把那个问题的答案或解决方案插入应用程序，然后桥梁更坚固，火箭更快之类的。你关心的原因是数学正在建立一套基本的解决问题工具，这些未解决的问题暴露了这些工具的差距和无能，以至于为了解决这些问题，想必你在制造更好的工具，而那些工具，思维工具，有时是字面工具，在其他地方有用。如果你不理解导致那个结果的轨迹，你就没有得到解决那个问题的全部期望好处，即知道创造了什么新工具来解决那个问题，我们现在可以用这些工具去解决其他问题。

Well, I think I think it does bear on truth a little bit in that if So, this is a a big topic in the nature of discussions people have is highlighting how to solve these problems is almost beside the point and often what you care about is the new ideas created in order to solve it and then there's spillover effects from those new ideas, which means simply seeing the solution is often not nearly as valuable as understanding the trajectory of thought that led to that solution. And a relevant part of that trajectory of thought is the body of work that you're building on and and where where that's important is you say like why do we care about, you know, solving this Navier-Stokes problem or like finding non-sophic groups or Jacobian conjecture, whatever one of these things is. Almost never are you plugging the answer to that question or the solution to that question into an application and then bridges are stronger and rockets go faster or anything like that. Like the reason you care is that math is building up like a fundamental fundamental set of problem solving tools, and these outstanding problems expose gaps and inabilities in those tools, such that in order to solve those problems, presumably you're making better tools, and those tools, tools of thought, sometimes literal tools, are useful elsewhere. And if you don't understand the trajectory of what led to that result, you're not getting out the whole like desired benefit of solving that problem, which is to know what new tools were created that solved that that we can now go and solve other problems with.

所以，除了出于礼貌的引用之外，如果你想知道世界是如何运作的，或者这些深层的数学真理是什么，忽视前人的思想史，我认为就是剥夺了自己对底层数学的理解。

And so, aside from that attribution just as good courtesy, simply if you want to know how does the world work, or what are these deep mathematical truths, to blind yourself to that preceding history of ideas is to rob yourself of an understanding of the underlying math, I think.

这似乎是个好时机来问问你的博客。你最近在陶哲轩的博客上写了一篇客座文章。很多人都读过。连我都读了，Grant。你能告诉我你写了什么吗？这个想法和概念是什么，以及你试图表达的一些关于我们在哪里、我们要去哪里、我们可能能做什么的思考？

This feels like a good point to ask you about your blog. You wrote a guest blog on Terry Tao's blog recently. Lots of people have read it. Even I've read it, Grant. Can you tell me about what you wrote? What was this idea and notion and some of the thoughts you were trying to put out there about where we are, where we're going, what we could maybe do?

是的，这是我想补充更多内容的东西，因为我认为在某种意义上我是在针对一个特定的语境，但阐述更广泛的背景是相关的。我写了，基本上，我要说的要点是，如果数学不仅仅是证明，我们应该承认那是什么，但然后我们应该庆祝那另一部分。另一部分，这不是唯一的一部分，但另一部分我非常珍视，就是，嗯，没有更好的词，让我们称之为解释。我用了“有动机的解释”这个短语，但有时在一个问题被解决和真正深入骨髓地感觉你理解了它之间有一个差距。这，我在这里有偏见，对吧？因为这就是我在制作 YouTube 视频时所做的一切，就是拿已经解决的问题，但试图说你怎么能自己发现这个？你知道，普通观众怎么能真正消化这个，并感觉不仅仅是我跟着那个证明的步骤，而是那种更深层的理解感。我想提出一个论点，即使在普及化和你我所处的世界之外，即使在研究的前沿，创造不仅仅是一个解释，而是我们称之为有动机的解释的那一面。每一步你都觉得你理解它来自哪里。每个新定义你都觉得你知道它为什么存在。我认为有可能为这样的定义创造一个定义，这样你可以因此获得学术信用，就像我们历史上给予证明学术信用一样。所以，如果今天决定一篇论文质量的大部分，或者直到最近，是它所证明的新结果的重要性，我认为一个同样重要的事情应该被衡量，应该被承认，是它为那个证明的具体步骤、为它创造的具体构造增加清晰度的程度，并试图达到那种更广泛的理解。

Yeah, it's something I want to add a lot more to because I think in some sense I was speaking to a specific context, but laying out the broader context is relevant. I wrote basically, the upshot of what I was saying is if math is something beyond just proof alone, we should acknowledge what that is, but then we should celebrate that other part. One other part, this is not the only one, but one other part that's very near and dear to my heart, is, well, for lack of a better word, let's call it explanation. I used the phrase motivated explanation, but there's sometimes a gap between a problem being solved and then a problem really deep in your bones feeling like something that you understand. And this is, I have a bias here, right? Because this is like all I engage with in making YouTube videos is taking problems that have been solved, but trying to say how could you have discovered this yourself? You know, how can a general viewer really digest this and feel not just I followed the steps of that proof, but whatever that deeper to your bones sense of understanding it is. I wanted to make the case that even outside of the world of popularization and where you and I live, even in just forefront of research, that side of creating not just an explanation, but what we'll call a motivated explanation. Each step you feel like you understand where it came from. Each new definition you feel like you know why it exists. I think that's possible to create a definition for such that you could get academic credit for that the same way we have historically given academic credit to proof. And so, if a lot of what determines the quality of a paper today, or up until quite recently, was the significance of the new results that it was proving, I think an equally important thing that should be measured, should be acknowledged, is the extent to which it is adding clarity for the specific steps of that proof, the specific constructs that were created for it, and trying to get at that broader understanding.

这里的背景是，你有一群数学家写了很多很多东西，经常使用“人类理解”这个短语作为数学的基本目标。这可以追溯到，如果不是更早的话，很多人都在呼应 Bill Thurston 1994 年的一篇论文，叫做《论证明与进步》，这很棒。每个人都应该读它。它在数学界享有盛名。所以，Bill Thurston 不是在回应 AI，而是在回应他认为的对数学的误解，即数学就像一个生产证明的工厂。他说：“那不是全部和最终。对于一个数学家推动领域前进来说，实际上还有更多。”他说：“如果有一个短语似乎真正抓住了这一点，那就是我们的目标是促进人类对领域的理解。”他正在为拓宽我们所说的意思而辩护。所以，是的，我把 Terry Tao 的提示当作一个机会来表达一些与 AI 无关但对我来说非常珍贵的东西，那就是我认为澄清一个结果的那一面，即使在它被证明之后，也值得信用，值得学术信用，因为我认为这往往是数学家觉得像是一个额外的附加物。你知道，他们就像，“哦，这不是我的严肃工作，但我还是想做。”所以，我在那里强调了一些具体的例子，比如 Timothy Gowers 在获得菲尔兹奖后所做的工作，他在你的播客中谈到了，或者我非常喜欢 Timothy Chow 的一篇论文，关于一个叫做 forcing 的概念，这以难以理解而闻名。在那篇论文的开头段落中，他说：“我们都熟悉开放研究问题。我想提出开放阐述问题的想法，即尽管它已被证明，但我们作为一个社区觉得它缺乏一个好的解释。”

The context here is that you have a bunch of mathematicians who have written many, many things often using this phrase human understanding as like the fundamental goal of math. And this dates back to if not much earlier, like a lot of people are echoing a 1994 paper by Bill Thurston called On Proofs and Progress, which is great. Everyone should read it. It is deservedly famous among the mathematics communities. So, Bill Thurston was not responding to AI, but he was responding to what he saw as a misconception of what math is as being like a factory that puts out proofs. He's "That's not the be-all and end-all. There's actually a lot more to what it means for a mathematician to push forward the field." And he's like, "If there's one phrase that seems to actually get it, it would be that our goal is to further human understanding of the field." And he was making case for broadening what we even mean by that. And so, yeah, I took the prompt from Terry Tao kind of as a chance to articulate something that is near and dear to my heart regardless of AI, which is that I think that side of clarifying a result even after it's been proven deserves credit, deserves academic credit, because I think it's often something that mathematicians feel is like an extra a little add-on. You know, they're like, "Oh, this isn't my serious work, but I kind of want to do it anyway." And so, I highlighted some specific examples in there, like work that Timothy Gowers did after his Fields Medal that he talked about on a podcast with you, or a paper I really like from Timothy Chow, about a concept called forcing, which is famously hard to understand. And in the opening paragraph of that, he says, "We are all familiar with open research problems. I want to submit the idea of an open exposition problem, which is something that, although it's proven, we as a community feel it lacks a good explanation."

你是说，数学家在基本的专业层面上，需要像发现者和探索者一样，也是解释者吗？

Are you saying that mathematicians at a fundamental professional level need to be explainers as much as they are discoverers and explorers?

是的，我是。而且澄清一下，我不是在谈论普及化。你有这种现象，在最前沿，你可以有一个小众中的小众中的小众，世界上只有不到 10 个专家真正理解一个特定的新结果。如果所有的注意力都花在进一步钻入那个洞，说哪些新结果建立在它之上，而没有同样的注意力让那些专家转向外，甚至向他们的邻居解释到底发生了什么。这感觉像是破坏了数学研究的更广泛目标，即促进人类理解。

Yeah, I do. And to be clear, I'm not talking about popularization. You have this phenomenon where at the very forefront, you can have a niche within a niche within a niche where there's like less than 10 experts in the world that really understand a particular new result. And if all of the mind share is spent on drilling even further into that hole and saying what new results do build on top of it, where there's not the same mind share for those experts to turn outward and explain even to their neighbors in a niche what exactly is going on. That feels like that undermines the broader goal of mathematics research, which is to further human understanding.

所以，他们不必是向公众解释的人。他们必须向同行数学家和同行专家解释。

So, they don't have to be explainers to the general public. They have to be explainers to fellow mathematicians and fellow experts.

还有同行科学家，对吧？我在帖子中想明确的另一点是，我们这样做。这已经发生了。这不是新事物。数学家们用他们的时间这样做，但我得到的印象是，这总是感觉二等。总是感觉像是他们所做的二等版本，与证明新结果相比。帖子的基本论点是，看，你已经在这样做了。我们都承认这是关于人类理解。我认为那种工作的地位，从根本上说是拿已证明的东西，但向外看，而且不必向整个世界，只是向你的邻近学者，值得同样的信用。那么，具体来说这意味着什么？我提出的一个事情是，如果有开放阐述问题的千禧年大奖问题会是什么样子，对吧？如果具体阐明社区认为，即使我们已经证明了这些，我们认为我们对这个没有足够好的理解，然后分配奖项给它。或者让它成为获得菲尔兹奖的一个贡献因素。就像新证明结果得到的所有东西一样。

And fellow scientists, right? And the other point I wanted to make clear in the post is like, we do this. This is what already happens. This is not like a new thing. Mathematicians do this with their time, but the impression I get is that that always feels second class. It always feels like a second-tier version of what they're doing compared to proving new results. And the basic case of the post is like, look, you're doing this already. We all acknowledge this is about human understanding. I think that the status of that kind of work that fundamentally is taking proven stuff, but looking outward, and it doesn't have to be to the whole world, just to your neighboring academics, deserves the same credit. And so, what would that mean concretely? One of the things I threw out there, what would it look like to have Millennium Prize problems for open exposition problems, right? What would it look like to have specifically articulated things that the community thinks, even though we've proven these, we don't think we have a good enough understanding of this. And then assign prizes to it. Or let that be a contributing factor to what gets you a Fields Medal. Like, all the same stuff that new proof results get.

还有加薪、工作机会之类的。

And pay rises and jobs and things like that.

没错，对吧？比如在决定终身教职的时候。所有这些，我觉得基本上我们已经在做了。它有一定的地位，但关键是提升它的地位，我认为。

Exactly, right? Like, in deciding tenure decisions. All of that, I think basically just we do this already. It has some status, but the case is to elevate its status, I think.

好的。我们现在可以谈谈批评意见了吗？

Okay. Can we do the criticisms now?

可以，可以，可以。我是说，那么，比如

Yeah, yeah, yeah. I mean, so I mean, what like

你想先来吗？

Do you want to go first?

我觉得有两个很容易提出的批评或反驳点。一个可能是，你会这么说，Grant。比如，如果

I think there's two quite easy criticisms to make or bits to push back. One would be like, you would say this, Grant. Like, if

因为你是个职业解释者。但公平地说，你谈论的不是你做的那种工作。你是科普者。你谈论的是

Because you're a professional explainer. But to be fair, you're not talking about the kind of work you do. You're a popularizer. You're talking about

是的，好吧。是的。我确实认为数学家应该更经常上 Numberphile，对吧？我很乐意，你知道，亲自和他们合作，但是，是的。但是，嗯，是的，所以，我并不想抬高我自己的工作。呃，但是，你知道，这可能是偏见。可能是潜意识的。我当然会关心那个。

Yeah, okay. Yeah. I do think mathematicians should go on Numberphile more often, right? I would love to, you know, work with them myself to like but yeah. But um Yeah, so it's I I don't Well, I don't mean to like elevate my own work. Uh but it's you know, it's probably a bias. That's probably subconscious. Like of course I would care about that.

好的。

Okay.

我看到的另一个常见评论很有道理。基本上是说：“好吧，当然，但是当模型在解释方面也超人，在证明方面也超人时，会发生什么？”嗯，那只是

The other frequent comment I saw makes abundant sense. It was basically like, "Okay, sure, but like what happens when models are superhuman at explanation as well as being superhuman at proof?" Um and that's just

是的。那是我的一个观点。在我的清单上。我觉得我们已经接近一个点，它们非常擅长解释东西和简化事物。

Yes. That was one of mine. That was on my list. I feel like we're already getting very close to a point where they are very good at explaining stuff and simplifying things.

有两种方式回应这个问题。一种是它们现在是否真的更好，或者它们更好会是什么样子，以及我们认为多久会出现超人的解释。我认为另一种途径是，当然，是的。那可能会发生。也许人们认为已经发生了。我不认为这否定了观点。我认为如果你说数学作为一个整体机构应该输出的不仅是问题的解决方案，还有解决方案来源的高质量解释，推动人类理解，无论是模型创造的还是学者创造的，我认为那仍然应该是被衡量和重视的部分。

There's two ways to engage with that question. One is whether they actually are better now or what would it look like for them to be better and how soon do we think it'll be the case before we have superhuman explanation. I think the other avenue is like E- Sure, yeah. That'll probably happen. Maybe people think that already has happened. I don't think that negates the point. I think if you say like math as a whole institution should be outputting not just solutions to problems but high-quality explanations of where the solutions came from that push forward human understanding, whether it's the model creating that or the academic creating that, I think that should still be a measured and valued part of what's coming out of it.

这确实引出了一个问题：如果 AI 能做证明和出色的解释，数学家的意义是什么？还剩下什么给数学家做？

It does beg the question that what is the point of a mathematician if the AI can do the proof and the brilliant explanation, what's left for the mathematician to do?

实际上，让我们来探讨一下。我认为那是更有趣的一半，即“你能做的任何事，我都能做得更好”的未来，思考一下。好吧，今年他们开始擅长解决数学家无法解决的问题。当然不是全部，但似乎有理由相信你会看到大量人们认为重要的新证明。好吧，所以证明生成是数学家工作的一部分。让我们试着列举数学家做什么，只是为了把它说出来。我在一个 Polymath 视频上听到一句很好的话，另一个数学频道，他们在描述其中一个结果时。那个人说：“好的数学家写证明，伟大的数学家写猜想，但最伟大的数学家写定义。”所以，我喜欢这个。你知道，除了你证明什么问题，简单地提出一个好问题。所以，Paul Erdős 可能属于这一类。他是个伟大的提问者。他真的有一种品味，找到正确的猜想，如果你能回答那个，就表明我们理解的有意义的进步。然后你有谁在写新定义。所以，这里有像 Galois 或 Grothendieck 或现代的 Peter Scholze，如果你说，“这些领域真正纪念碑式的人物在创造什么？”不是他们解决的任何一个问题，而是像一种全新的思考方式，通过一套新的定义来代表。所以，过去一个世纪数学的转变之一是转向以所谓的范畴来思考，如果你试着问数学家，“范畴解决了什么新问题？”他们可能能找到一些，但你会实际上很难得到好的答案，关于因为我们对范畴论的理解而解决的开放问题。相反，更像是那是一种重新构建人们思考方式的整个方式，提供了对许多不同事物的更清晰理解，并使他们的思维更迅速地从一个领域转到另一个领域并识别联系。我不知道 Peter Scholze 到底在研究什么，但那些似乎深入其中的人，都是关于找到拓扑的正确定义。他说，我们有点没有正确定义它，但让我们正确定义它。嗯，你有纲领设定，比如 Langlands 纲领，不是特定的开放问题，而是更多一大群可以表述为问题的问题，如果这些都被解决，就会揭示许多不同领域之间的广泛联系。所以纲领设定是某种非常高层次的东西，但它是数学家工作的重要部分，甚至推动它前进。所以，我认为任何时候你在讨论数学家现在做什么，你都在转移焦点，也许好吧证明可以由机器生成，所以转而关注应该证明什么。一旦他们擅长成为好的提问者，然后你说也许转向定义。也许转向纲领设定。也许转向解释。科幻式的探讨是说每一个部分都是，你知道，你有一个全知的神一样的存在。数学家能做的任何事，它都能做得更好。嗯，即使在这种极端情况下，我从朋友那里听到的一件事，我很喜欢，是也许未来——好吧，我说喜欢不是指这是我们所有人都想要的未来，而是它似乎是一种新颖的描述方式。也许数学家的未来是他们像博物馆馆长一样行事，那里有大量的内容在博物馆里或可能放在博物馆里的东西，但必须有人做出选择，这是值得被走过博物馆的人看的东西。

And actually, let's engage with that. I think that's the more interesting half to engage with where this is the like anything you can do, I can do better future to think about on Okay, this year they are starting to get good at solving problems that mathematicians couldn't solve. Not all of them, certainly, but like it seems there's cause to believe that you're going to see a flood of new proofs of results that people find significant. Okay, so proof generation is one component of what mathematicians do. Let's just try to enumerate what it is mathematicians do just to like get it out there. There's one nice quote that I heard on a Polymath video, another math channel, as they were describing one of these results. The person said, "Good mathematicians write proofs, great mathematicians write conjectures, but the greatest mathematicians write definitions." So, I like that. You know, aside from what problems are you proving, simply asking a good question. So, Paul Erdős would maybe fall in this category. He was a great question asker. He really had a taste for finding what's the right conjecture such that if you could answer that, it indicates meaningful progress in our understanding. And then you have who's writing the new definitions. And so, here you have it's like Galois or Grothendieck or in modern times Peter Scholze, where if you say, "What is it that these like truly monumental figures in the field were creating?" It's not so much any one problem that they were solving, it was like a completely new way of thinking about a body of work as represented by a new set of definitions, essentially. And so, one of the shifts in math over the last century is one towards thinking in terms of something called categories, and that it's if you try to ask, "What new problems do categories solve?" to a mathematician, they might be able to find some, but like you'll actually struggle to get good answers on, you know, what's the open problem that is solved because of our understanding of category theory. Instead, it's more like that was an entire way of reframing how people think that provided a cleaner understanding of lots of different stuff, and it made it more swift for their brains to go from one field to another and recognize connections. I don't know what the heck Peter Scholze is working on, but those who seem to be like deeply into it, it's all around like finding the right definition of topology. And he's like, we sort of we haven't defined it correctly, but let's define it correctly. Um you have program setting, so like the Langlands program as like not a specific open problem, but more a large body of what could be articulated as problems like that were these all to be solved expose like a broad connection between many different fields. And so program setting is like some very high-level thing, but it's an important part of what mathematicians do to even push it forward. So, I think anytime you're having these discussions on what your mathematicians do now where you're shifting the focus maybe okay proofs can be generated by machines, so instead focus on asking what should be proven. As soon as they get good at being good question askers, then you say maybe you shift towards definitions. Maybe you shift towards program setting. Maybe you shift towards explanation. The sci-fi engagement is to say every single part of that is, you know, you've got some all-knowing god-like entity. Anything the mathematician can do, it can do better. Um even in that extreme, the one thing I heard from a friend uh that I quite liked was that maybe the future Okay, I say I like not in the sense of this is a future that we all want, but just it seemed like a novel way of describing things. Maybe the future of mathematicians is that they act something like museum curators where there's a huge body of contents in that museum or things that could be in a museum, but someone has to make the choice this is what deserves to be looked at by the people walking through the museums.

就像卢浮宫储藏室里的所有东西，但你看不到。是的。

Like all the stuff that's in storage at the Louvre, but you don't see it. Yeah.

对。或者 Brady 壁橱里的所有东西。

Right. Or everything that's in the Brady's closet.

就像你在决定你身后架子上放什么，对吧？而且我敢肯定，有一个巨大的存储设施，里面全是可能放在那里的候选品。

Like some you're deciding what sits on that shelf behind you, right? And what I'm sure there's a whole storage facility of potential candidates on what sits there.

确实有一个存储单元。

There is a storage unit.

我毫不怀疑。所以，我确信数学可能包含的内容如此庞大，以至于需要有人来基本决定哪些东西值得我们人类关注。你可能会说，为什么机器不能做这个？你只需问机器我该看哪些。所以，在任何超级 AI 包办一切的未来中，最安全的东西我认为是那些关系性的，那些依赖于人与人之间的关系从而产生价值的东西。这不仅仅是屏幕上的像素。因此，整个博物馆策展人的形象在某种程度上承认了这不仅仅是客观的，你知道，这是博物馆里最好的展示，因为你可以给它赋予某种数量。而是因为对那个人的信任，或者他们在领域中的地位，或者他们过去的成就。

I have no doubts. And so, what I'm sure of the body of what math can be is so huge that having someone basically decide which of these things deserve our like humans our attention. I think you could say well why can't the machines do that? You just ask the machine which of these like should I look at. So the stuff that ends up being safest I think in any sort of like super you know AI is doing everything style future are the things that are relational the things that hinge on um it's a a human's relationship with another human that get that causes the value to come about. It's not just what are the pixels on the screen. And so I think the the whole museum curator image there it's somehow acknowledging that it's not just an objective you know this was the best thing to show in the museum because of whatever quantity you could put to it. It's because of a trust in that individual or the the status they have in the field or what they've had.

读完你的博客后，我还有一个批评。这不是批评，是个问题。因为你勾勒出了这个新的乌托邦，数学家根据他们的解释来评判。他们的有动机的解释，我们据此决定谁获得教授职位和终身教职。奖项也是这样颁发的。我们新的千禧年问题变成了解释问题而不是证明问题。我的问题是，这听起来有点模糊，缺乏严谨性。比如，我们如何判断某件事被很好地解释了？因为对一个人来说解释得很好的东西，对另一个人来说可能解释得很差。这就变成了非常主观的，‘哦，你看到那个解释了吗？太棒了。’而我会说，我觉得那解释得一点都不好。而证明就是证明就是证明。它要么真要么假。这似乎是数学的支柱之一：严谨、真理和正确性。而你似乎漂移到了一个更注重理解和解释、确保人们认同的领域。我只是不知道你如何用任何形式的严谨性来衡量这个。

I have one remaining criticism after reading your blog. It's not a criticism, it's a question. Because you sort of set out this the new the your new utopia where mathematicians are judged on their explanations. Their motivated explanations and that's how we decide who gets the professorships and tenure. That's how that's how prizes are given. That's what our new our new set of millennium problems become explaining problems rather than proof problems. My problem with that is it sounds kind of a bit bit wishy-washy and lacking in rigor. Like how do we decide that something has been well explained? Because something well explained to one person is something poorly explained to someone else. And it just becomes this very like subjective oh have you have you seen that explanation? It's brilliant. And I'll be like I don't think that was well explained at all. Whereas a proof is a proof is a proof. It's true or it's not true. And that seems to almost be one of the backbones of mathematics is rigor and truth and rightness. And you seem to be drifting into a lane where what matters more is understanding and explaining and making sure my, you know, these people are on board. And I just don't know how you would measure that with any form of rigor.

我认为，我之所以强调‘有动机的’这个形容词，说我们应该衡量或提升有动机的解释的地位，而不是仅仅解释或清晰解释或去神秘化之类的。我认为这更接近一种可验证的属性。这更接近你可以制定评分标准的东西。我这么认为的原因是，当我制作视频时，我实际上会想，‘嘿，这一步，观众有理由相信我们正在迈出这一步吗？’你几乎可以像清单一样过一遍。不过你说得对，这是主观的。我认为，试图为开放阐述设立类似千禧年奖问题的简单失败模式是，写出来的东西并不明确，比如，‘嘿，这符合评分标准吗？’我认为这必然是任何涉及‘人类理解’这个短语的事情的一部分。所有数学家都在重复这个短语，我们正在推进人类对它的理解。你说，‘好吧，理解的人性，与那种人性联系在一起，将是一种主观性，这需要做其他领域所做的事情，对吧？在衡量进展时，它不会有证明那样超级严格的可检验性。但是，你仍然可以围绕有点主观的事情制定评分标准。就像教育工作者给非数学作业评分时，仍然能够有相当标准化的，比如，满分九分。在这篇论文上得七分而不是八分意味着什么？因为可以在其他领域制定评分标准，或者因为法律体系存在，对吧？法律体系是这样一个体系，所有那些措辞都试图以数学那样的严谨和精确来达到某种东西，但它是在一套人类规范之内，永远不可能有那种严谨。所以，我认为在开始所有这些之前，你必须实现的目标之一是，社区就成功的定义达成一致。现实地说，可能只是一个由人组成的小组，用他们最好的判断来判断这样一个开放阐述问题的精神是否得到解决。我认为永远不会有一条清晰的界线，客观地，你把它输入电脑，电脑输出说，是的，确实一切都有动机。我想提出的一个理由是，值得一试。值得承认的是，如果你认真对待‘人类理解’这个短语，那就意味着要参与与之相关的主观性。其他领域这样做。数学没有理由不这样做。

I think So, one of the reasons I I leaned on the um adjective motivated and saying like we should measure uh or we should elevate the status of motivated explanation as opposed to just explanation or lucid explanation or demystifying or something. I think that's closer to being a verifiable property. That's closer to being something that you could put a rubric for where and the reason I think that is cuz when I put together videos, I'll actually think like, "Hey, this step, is there a reason that the viewer would believe we're making this step?" You can almost like go through as a checklist to say. Um you are right though, it is subjective. I think the easy uh failure mode of trying to say something like a Millennium Prize problem for open exposition would be that a thing is written and it's just not clear-cut like, "Hey, does this does this satisfy the rubric or not?" I think that's necessarily part of anything that engages with uh this phrase, you know, human understanding. All the mathematicians are repeating the phrase we're advancing human understanding of it. You say, "Okay, that humanness of the understanding tied with that humanness is going to be a subjectivity uh that that requires doing what every other field does, right? Um in terms of when they measure progress, it's not going to have the same super rigorous checkability of proof. But, you can still write rubrics around things that have a little bit of subjectivity to them. In the same way that I don't educators writing homework that's not math are still able to have like a pretty standardized, you know, grade this out of nine points. What does it mean to get seven instead of eight on this essay? Because it's possible to write rubrics in other domains or because the legal system exists, right? And then the legal system is one where all of that phrasing is trying to get at something with the same rigor and exactness that say math does, but it's in inside a set of human norms that could never possibly have that. Uh so one of the I don't know. Uh Uh goals that I think you would have to uh accomplish before you could even start all of that would be something that the community agrees on as a definition of, you know, success here. Realistically, probably would just be a panel of people who uh are using their best judgment for whether the spirit of such an open exposition problem like has been addressed. I think there's never going to be such a clear-cut line where in an objective, you know, you plug it into a computer that can come out on the other and say like, yes, indeed everything was motivated. Um I think one of the cases I want to make there is like, it's worth trying. It's worth acknowledging that if you're serious about this phrase human understanding, that means engaging with the subjectivity uh associated with that. Other fields do this. There's no reason that math shouldn't as well.

Grant，我知道你是个解释者，对吧？但你不也有点兴奋于那些终点线和证明吗？比如黎曼假设、纳维-斯托克斯方程、费马大定理，这些有明确终点线的东西。作为一个非数学家，我觉得这是数学最酷的事情之一。我喜欢它。它感觉像是发现和前沿之类的，我不想失去这个。我觉得这很酷。

Grant, aren't you I know you're an explainer guy, right? But aren't you a little bit excited about the finish lines and the proofs that you know, your Riemann hypothesis, your Navier-Stokes, your Fermat's Last Theorem, these things that have these clear clear like like I feel like that's one of the coolest things about maths for me as a non-mathematician. I think it's like I love it. It feels like discovery and frontier and that sort of stuff and and I don't want to lose that. I think that's cool.

为什么会失去呢？你仍然会有黎曼假设被解决的那一刻。它会从没有证明变成有证明。如果我们用 AI 来做，一旦有了证明，可能没有人能完全理解它，至少在证明被创造的那一刻。但是，你仍然会有那个终点线。你觉得那个终点线不再有过去的光泽和光辉了吗？或者你觉得我们应该仍然以同样的热情庆祝它，不管是谁跨越了它？

Why is that lost? Like you still you still have the moment when Riemann hypothesis will be solved. It'll go from not having a proof to having a proof. Probably if we if it's done with AI, once it has a proof no no one will understand it fully, at least the moment that the proof is created. But, you'll still have that finish line. Um do you feel that do you feel that finish line no longer has the the luster and the glow that it used to? Um Or or or do you feel like we should still be celebrating that with the same enthusiasm that we did before, regardless of the entity who's crossing it?

它仍然有光泽和光辉，对吧？就像我仍然喜欢这些浪漫的问题，它们困扰了我们几十年。它会缺少人类的故事。它会缺少安德鲁·怀尔斯把自己锁在阁楼里，保守一切秘密，然后有一天出现在黑板前说，‘看这个。’现在，一切都会，现在，总是会是一样的。

It has It still has the luster and the glow, right? Like I still I still like the idea of these romantic problems that have baffled us for you know, decades. It It will lack the human story. It will lack Andrew Wiles locking himself in an attic and keeping everything a secret and then one day appearing in front of a blackboard and saying, "Look at this." Now, it's all going to Now, it's always going to be the same thing.

它会被 AI 用 PDF 吐出来，那样就缺少了那种浪漫。它仍然有光泽。但我想问的是，随着前沿不断推进，数学的前沿会不会变得超出人类的理解范围？就像我对我 4 岁的儿子讲微积分，他根本听不懂，他做不到。但 20 年或 10 年后，他就能理解了。可是，我们是否面临这样的危险：数学的前沿已经远远超前，以至于没有任何人类能够企及？超级智能正在创造猜想和证明，而人类永远无法理解，因为我们没有那个硬件。

It's going to be spat out by a PDF by an AI, which will lack that. It will lack the romance. It still has luster. But I guess a question I have is, as the frontier moves further and further, is there a danger of the frontier of mathematics becoming beyond human comprehension? Like, if I took my 4-year-old boy and tried to talk to him about calculus, he's just not going to get there. He can't do it. And then in 20 years, or 10 years, or whatever, he will get there. But are we in danger of the frontier of mathematics getting so far ahead that no human can ever get there? And the superintelligence is creating conjectures and proofs and things that will never be comprehended by a human, because we haven't got the hardware.

这似乎很可能。还有很多其他领域。我想经济学家会写到，没有人真正理解铅笔是如何制造的。它是存在于任何个体头脑之外的东西。或者没有人理解 iPhone 是如何制造的，意思是如果你让任何一个人写下来，没有人能做到。现在很多数学已经超出了——当然，数学作为一个整体早已超出任何人的理解。任何具体的数学结果，有相当多只有世界上不到 100 人甚至不到 10 人能理解。从实用意义上说，这也可能是我同意围绕费马大定理或黎曼假设被解决时有某种光泽的原因。但我自己与数学的关系感觉——我找不到更好的词——更接近一种宗教体验，或者更接近人们谈论精神存在的方式，它都围绕着我感觉自己理解的东西，以及那种理解到来的时刻。所以这就是我。只是 Grant Sanderson 的偏见。我认为我可能没有对做研究产生同样的吸引力，而是被现在做的事情所吸引，其中一个原因是我个人在把一个结果——即使它已经已知几百年——从我不理解、即使读了证明也感到晦涩，到坐下来花时间努力理解，然后达到那种豁然开朗的时刻中，找到更多快乐。即使不是提出新东西，那种满足感和对生活的深刻丰富，让我对那一面更有感情，而不是听到某个新结果被证明。现在，我认为你完全正确的地方是，你不会得到同样的安德鲁·怀尔斯式的故事，没有人能拍一部关于某个人坐下来一个下午就感觉自己真正理解了费马大定理之类的漂亮纪录片。但就我自己对这个领域的热爱而言，如果我内省什么更打动我——是那种从不懂到懂的豁然开朗时刻，还是我听说黎曼假设被解决了？它完全属于前者，这可能就是为什么我——再次，我知道可能不是每个人都分享这种感觉，而且如果绝对前沿不仅对大多数人、不仅对除了小众之外的所有人，而是对所有人都不可及，那确实会失去一些东西。但从个人角度来看，前沿对所有人不可及的世界，与当前前沿对几乎所有人不可及的世界，没有什么不同。

That seems likely. And there are lots of other domains. I think economists will write about how nobody truly understands how a pencil is made. It's this thing that exists outside of any one individual's head. Or like nobody understands how an iPhone is made, in the sense that if you ask any individual to write it down, no individual will be able to. A lot of math now is beyond—certainly math as a whole has been beyond anybody's understanding for a long time. Any specific mathematical result, there are quite a few that less than 100 people in the world or even less than 10 people in the world understand. In a pragmatic sense, and this is maybe also why I agree there is a luster around when Fermat's Last Theorem is solved or when the Riemann hypothesis is solved. But where my own relationship with math feels—I don't know, for lack of a better word—closer to a religious experience or closer to the way people talk about their spiritual existence, it's all sited around stuff that I feel like I understand and what that moment of coming to understand felt like. And so this is me. It's just Grant Sanderson biases. One of the reasons I think I maybe didn't feel the same draw towards doing research and felt a draw towards doing what I'm doing now is that I personally find a lot more joy in taking a result, even if it's been known for a couple hundred years, that I didn't understand, that felt opaque even after reading the proof, sitting down with it for a while, trying to wrap my head around it, and then coming to that clicking moment. Even if it's not coming up with something new, that is such a satisfying feeling and such a deeply enriching part of life to me that I just feel more affection for that side of it than if I hear some new result has been proven. Now, I think where you are totally right is you don't get the same kind of Andrew Wiles type stories, and no one's going to be able to make a nice documentary about when one individual sat down for an afternoon and felt like you really understood Fermat's Last Theorem or something like that. But as far as where my own love of the field comes from, if I'm being introspective about what moves me more—is it that clicking moment of feeling like I went from not understanding to understanding, or I hear that the Riemann hypothesis has been solved? It's just so squarely in the former category, which is maybe why I—and again, I know maybe not everyone shares that feeling, and there is something that would be lost if the absolute forefront is inaccessible not just to most people, not just to all but the niche, but instead it's inaccessible to everyone. But from an individual standpoint, that world of the forefront being inaccessible to everyone is no different from the present world of the forefront being inaccessible to almost everyone.

大多数领域，我认为科学和数学也是如此，都受益于英雄，比如英雄故事。我担心我们会用完让人们成为英雄的机会，让人们成为安德鲁·怀尔斯、诺贝尔奖得主、菲尔兹奖得主、陶哲轩，这些人——

Most fields of endeavor, and I think it's true for science and mathematics as well, benefit from heroes, like hero stories. And I worry we're going to run out of chances for people to be heroes, for people to be Andrew Wiles, Nobel Prize winners, Fields Medalists, Terry Tao, these people that—

这个，你不是吗？就像——你曾论证过为什么诺贝尔奖应该总是归于一个或少数几个人。

This, haven't you? Like the—you made a case for why the Nobel Prize should always be down to one or a handful of individuals.

是的，而不是欧洲核子研究中心之类的。是的，是的。就像我认为有些——是的，我谈过这个。我认为但我觉得那些——我的意思是，你可以回来说：“好吧，也许英雄会是解释者。”也许你是对的。也许那会是人们成为英雄的唯一地方。

Yeah, rather than CERN or something. Yeah, yeah. Like I think there's something—yeah, I have spoken about it. I think but I do think like those—and I mean, you could come back to me and say, "Well, maybe the heroes will be the explainers." And maybe you're right. Maybe that would be the only place for people to be heroes.

我承认那没有同样的英雄气概。

I admit that that doesn't have the same heroism.

是的。是的。我只是担心，一旦我们开始从清单上划掉一些大问题，如果取而代之的新问题只属于计算机，我们就会用完英雄。当你用完英雄时，那会影响人们成为数学家、科学家或其他什么的动力吗？我不知道。

Yeah. Yeah. And I'm just worried once we start scratching a few of these big problems off the list, and if the new problems that take their place become computers only, we're going to run out of heroes. And when you run out of heroes, will that affect people's motivation to become mathematicians, scientists, whatever? I don't know.

我确实认为这是一个真实的担忧，那就是可能激励很多人进入这个领域的一部分，是他们年轻时眼中的那一丝光芒，想着如果我是解决那个问题的人呢。不是每个人，对吧？但我确实认为有相当数量的数学家，那可能就是推动他们前进的东西，尤其是在早期。

I actually do think that's a real worry, is that some of what might motivate a lot of people going into it is that glimmer in their eye as a young teenager of thinking what if I was the one to solve that problem. Not everyone, right? But I do actually think there's a meaningful number of mathematicians that that was probably what was pushing them along, especially in the early years.

嗯。

Mhm.

我不认为我们应该这样做，但假设保持英雄主义是我们希望这个领域拥有的重要部分，我认为有一种方法可以做到，那就是把竞赛数学的传统延续到成年，而不仅仅是给孩子们的东西。你会有一个像国际数学奥林匹克那样的东西，在世界上拥有与职业围棋或职业国际象棋锦标赛相同的影响力，它不只是给孩子们的，而几乎是一种观赏性运动，你惊叹于某些有天赋的问题解决者的原始能力，周围有热烈的气氛，有从上一届锦标赛中公认的最佳选手，人们有排名，我们像追体育一样追它。如果你想，我认为如果你想在数学中保持英雄主义，那可能是——我不会说那是唯一的方法，但如果我是世界之王，我只需要让那发生，那就是我会启动的项目：面向成年人的国际数学奥林匹克。我会用不同的方式表述。与刚才的措辞相比，我们需要一个更好的营销部门。

I don't think we should do this, but if let's say maintaining heroism was an important part of what we want the field to have, I think there's a way to do that, which is you continue the tradition of contest math into adulthood instead of just something that we give to the children. And you would have something like an international math olympiad that has the same presence in the world as professional go or professional chess tournaments have, that it's not just for the kids, but it's almost like a spectator sport where you marvel at the raw ability of a certain set of talented problem solvers, and there's a fanfare around it, there's someone who's known to be the best from that last tournament, people have rankings, and we follow it like a sport. If you wanted to, I think if you wanted to maintain heroism in math, that's probably—I'm not going to say that's the only way to do it, but if I was king of the world and I just needed to make that happen, that's the program I would start: international math olympiad for grown-ups. I would frame it differently. We'd need a better marketing department compared to the phrasing I just used.

早期阶段，早期阶段。我们会改进的，但我明白你的意思。不过，我最喜欢的英雄并不总是运动员或赢得比赛的人。

Early days, early days. We'll workshop it, but I hear you. But for my favorite heroes, though, are not always sports people or the people who win contests.

他们往往是你的尼尔·阿姆斯特朗、你的埃德蒙·希拉里，那些真正身处前沿的探索者。我希望我们不会失去他们。我希望我们不会失去他们。

They often are your Neil Armstrongs, your Edmund Hillarys, those people that actually are at frontiers and explorers. I hope we don't lose them. I hope we don't lose them.

那么，再看另一个例子。让我们从历史模式转向一个完全不同的方向。我认为我们可以看看两个例子：计算圆周率的位数，以及混沌理论。计算圆周率的位数，这曾经是人们手工做的事情。你甚至会有英雄。我不确定他在当时是否被英雄化，但马特·帕克已经纠正了这一点，威廉·尚克斯就是一个有点疯狂的家伙，他喜欢计算圆周率的位数，并且比任何人都算得更远。人们会根据数学的新进展想出新的策略。但在计算机时代，它开始运作的方式，一方面完全否定了世界上的威廉·尚克斯们。就像，计算机可以做他正在做的事情。但尽管如此，仍然有一个持续的传统，即最长的圆周率计算，现在转向成为一个计算问题，一个协调超级计算机的问题，一个提出更好算法的问题。甚至到今天，对吧？当人们计算出比以往更多的位数时，那是一个时刻，是一个人们关注的记录，即使它与人类手工计算有质的区别。也许在一个证明越来越多由机器生成的世界里，你会看到某种类似的质变，不再是某个人亲手或在自己的头脑中写出黎曼猜想的证明，而是他们想出了正确的方式来协调所有机器，从而比默认情况下更迅速地生成了整个程序或类似的东西。

Here's another case then. Let's take a totally different direction from patterns in history. I think two that we could look at would be calculating digits of pi, and then chaos theory. Where calculating digits of pi, this used to be something that people would do by hand. You would even have heroes in that. I'm not sure he was heroized in his time, but Matt Parker has corrected this, but William Shanks was one just kind of bonkers dude who just loved calculating digits of pi and did it much farther than anyone else would have. There were newer tactics that people would come up with based on newer advances in math of how to do this. But the way that it started to work in an age of computers, on the one hand, completely nullified the William Shankses of the world. Like, the computer can do what he was doing. But nevertheless, there was a continued tradition of what is the longest calculation of pi that now shifted towards being a computational question, one of orchestrating supercomputers, one of coming up with better algorithms there. And even to this day, right? When people have calculated out to even more digits than we had before, that's a moment, and it's a record that people look at, even if it has the qualitative difference from humans doing it by hand. Maybe in a world of proofs being increasingly generated by machines, you have some kind of similar qualitative shift where it's not so much who is the one who is writing by hand or in their own mind the proof of the Riemann hypothesis, but they've come up with the right way to orchestrate all of the machines such that they've generated an entire program or something like that more expeditiously than would have happened by default.

那么，什么，像 AI 提示者会成为新的国王吗？

So, what, like AI prompters will become the new kings?

是的，每次你用“提示者”这个词，都让人感到如此沮丧。

Yeah, every time you use the word prompter, it feels so depressing.

我知道。我知道。我知道。

I know. I know. I know.

但你知道，除了提示之外，还有真正的工程可以投入到这样的事情中。

But you know, there's aside from prompting, there's like real engineering that can go into things like that.

更好的词。我们需要一个更好的词。好吧。

Better word. We need a better word. All right.

我们需要一个更好的词。

We need a better word.

你可以从另一个角度看待这个问题，即计算机如何改变了做数学的方式？我认为混沌理论的起源非常有趣，因为这是一块数学，你可以在计算机存在之前就阐述并写出证明。但是，导致人们甚至提出这些定义的观察现象几乎需要计算机的存在，因为你会有一个系统，你给它一些初始条件，你会看到它如何发展，但然后你会运行这样的模拟 10,000 次，看看对初始条件的细微变化如何导致最终模型中相当不同的行为。这种方式你几乎无法手工完成，因为你需要运行成千上万次模拟，每一次都需要数百万次计算。因此，通过自动化计算，你达到了一个点，可以观察到世界上一种独特类型的现象，人们知道然后开始围绕它制定数学理论。所以，从这个角度，你可以说，就像计算从铅笔和纸到计算机导致了类型的差异，而不仅仅是数量的差异，因为能够进行数百万和数十亿次计算并观察到最适合用混沌理论阐述的模式。当你拥有丰富的证明写作时，它会是什么样子？是否有一种我们目前完全无法理解的元数学，但只能通过观察当机器在某个领域创建成千上万的证明时会发生什么，并且所有这些证明的特征显示出一些新的涌现现象，而当你只是手工一个接一个地写每个证明时，这些现象是无法观察到的，但现在，从这个更高层次的角度，你可以看到。因此，英雄将是相当于混沌理论早期开始认识到这是一组值得定义的现象。你可能会发现类似的英雄，他们拥有这种 10,000 英里高的证明视角，在某个领域，成千上万的证明被生成并暴露出某种模式，并意识到，“嗯，有一种正确的方式来理解所有这些模式中发生的事情，如果我们以更慢的速度进行所有这些，我们永远不会知道。”那将是一个不同的英雄主义空间，在一个勇敢的新世界中。

Another angle you could take on this in terms of how did computers change doing math? I think the origins of chaos theory are pretty interesting because here's a piece of math that you could articulate, you could write the proofs for before computers existed. But the observed phenomena that caused people to even come up with these definitions almost required computers to exist in that you'd have some system, you'd give it some initial conditions, you would see how it played out, but then you'd run a simulation like that 10,000 times and see how subtle changes to those initial conditions cause quite different behaviors in the ultimate model. In a way that you almost couldn't do by hand because you needed to run those thousands and thousands of simulations, each one of which requires millions and millions of computations. And so, by automating computation, you got to this point where a distinct type of phenomenon in the world could be observed and people knew then to start formulating a theory of math around that. So, from that angle, you could say in the same way that computations going from pencil and paper to computers caused a difference in type, not just a difference in amount for what the math was by virtue of being able to do millions and billions of computations and observing patterns that are best articulated with chaos theory. What does it look like when you have an abundance of proof writing? Is there some sort of meta mathematics completely unfathomable to us right now, but that can only come about by observing what happens when you have a machine go and create like whole swaths of tens of thousands of proofs in a certain field and the character of all of those proofs shows some new emergent phenomenon that would not have been observable when you're just writing each proof by hand one at a time, but now, from this higher-level perspective, you can see that. And so, the hero would be the equivalent of your early years of chaos theory starting to recognize this as a deserving set of phenomena, a set of phenomena that deserves a definition. You might find similar heroes who have this 10,000-mi high perspective of proof in some field as in tens of thousands of proofs being generated and exposing some kind of pattern and realizing, "Mhm, there's a correct way to understand what's going on with all those patterns that we never would have known had we gone slower with all of those." That would be a different room for heroism in a brave new world.

好了，今天就到这里。感谢格兰特抽出时间，请查看下方链接了解所有与桑德森相关的内容。我还将在下方链接到播客《信号与线程》。这是由我们的赞助商简街制作的。《信号与线程》充满了深入访谈，涉及与简街作为全球交易公司的核心业务相关的各种话题，这意味着你会听到一些最优秀的人从事机器学习和 AI 等工作。事实上，最近一集是关于 AI 时代教学的精彩对话。一定要看看。正如我所说，链接在下方。它既有音频也有视频。对我来说就到这里。我是布雷迪·哈兰，你一直在收听 Numberphile 播客。

Well, that's all for today. Thanks to Grant for his time, and do check out the links below for all things Sanderson-related. I'll also be linking down below to the podcast Signals and Threads. That's produced by our sponsor Jane Street. Signals and Threads is filled with in-depth interviews about all sorts of stuff tangentially related to Jane Street's core business as a global trading firm, and that means you hear from some of the very best people working on things like machine learning and AI. In fact, the most recent episode is a great chat all about teaching in the age of AI. Do check it out. As I said, link down below. It's available as both audio and as a video. That's all for now from me. I'm Brady Haran, and you've been listening to the Numberphile podcast.