What Is the Positive Grassmannian and Why Does It Show Up Everywhere?
What links certain mathematical models of traffic flow, shallow-water waves, and quantum particle scattering?
交通流、浅水波和量子粒子散射的某些数学模型有何联系？
The surprising answer lies in a corner of the algebraic combinatorics world that goes by the name of positive Grassmannian.
令人惊讶的答案就在代数组合世界的一个角落里，这个角落被称为正格拉斯曼函数。
In simple terms, the positive Grassmannian is a shape that classifies other shapes.
简单来说，正格拉斯曼是一种对其他形状进行分类的形状。
Remarkably, pieces of the positive Grassmannian can be reassembled in forms that reveal shared structures in these and many other seemingly unrelated mathematical systems.
值得注意的是，正格拉斯曼的各个部分可以以揭示这些和许多其他看似不相关的数学系统中的共享结构的形式重新组装。
That we know the positive Grassmannian crops up in many real-world settings is largely down to the theoretical work of Lauren Williams at Harvard University.
我们知道积极的格拉斯曼主义在许多现实世界中出现，这在很大程度上要归功于哈佛大学劳伦·威廉姆斯的理论工作。
In this latest episode of The Joy of Why , Williams talks to co-host Steven Strogatz about her work, how she realized the surprising pervasiveness of the positive Grassmannian, and how she has made a career of finding connections among fields that don’t at first sight seem connected.
在最新一集的《为什么的喜悦》中，威廉姆斯与联合主持人史蒂文·斯特罗加茨谈论了她的工作，她如何意识到积极的格拉斯曼主义令人惊讶的普遍性，以及她如何在乍一看似乎没有联系的领域之间寻找联系，从而开创了职业生涯。
The conversation then switches to another project Williams is working on, called First Proof, which is trying to measure objectively how good AI systems are at coming up with proofs of research-level mathematical statements, and which leads to an exploration of whether AI may or may not take over mathematics.
然后话题转到威廉姆斯正在进行的另一个项目，名为“First Proof”，该项目试图客观地衡量人工智能系统在提出研究级数学陈述证明方面的表现，并引发对人工智能是否可能接管数学的探索。
Listen on Apple Podcasts , Spotify , TuneIn or your favorite podcasting app, or you can stream it from Quanta .
在 Apple 播客、Spotify、TuneIn 或您最喜爱的播客应用程序上收听，或者您也可以从 Quanta 进行流式传输。
Note : Since this conversation was recorded, results from the First Proof Second Batch project were released on June 10, 2026 .
注：自本次对话录制以来，第一批第二批项目的结果已于 2026 年 6 月 10 日发布。
Okay, now we’re starting, starting?
好的，现在我们开始了，开始吗？
And this is The Joy of Why .
这就是“为什么的喜悦”。
A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today.
来自《Quanta》杂志的播客，我们探讨了当今数学和科学中一些尚未解答的最大问题。
I can start us off with a discussion I had recently with a mathematician who works in the area of math that we call algebraic combinatorics.
我可以从最近与一位数学家的讨论开始，这位数学家从事我们称为代数组合学的数学领域。
Yeah?
是的？
What makes you say “oof” to that?
是什么让你对此说“噢”？
Oh, I don’t know.
哦，我不知道。
It sounds really hard.
听起来真的很难。
I mean, algebraic, we know what those two words mean.
我的意思是，从代数角度来说，我们知道这两个词的含义。
Combinatorics, I assume some kind of discrete algebra.
组合学，我假设某种离散代数。
Yeah, something like that, right?
是的，类似的事情，对吧？
It’s all about discrete objects or structures.
这都是关于离散的对象或结构。
When I think of combinatorics, I sometimes think of the many ways in which you can organize something.
当我想到组合学时，有时我会想到组织事物的多种方式。
Like this comes up in thermodynamics where statistical thinking or in counting of entropy.
就像热力学中统计思维或熵计数中出现的情况一样。
What are the possible ways of organizing a sequence of numbers, for instance?
例如，组织数字序列的可能方法是什么？
And is that an element in combinatorics, or is that too simplistic?
这是组合学中的一个元素，还是太简单化了？
No, that’s exactly on the money.
不，这完全是为了钱。
That’s the kind of thing that we’re thinking about.
这就是我们正在考虑的事情。
And for our listeners who may not be in the world of thermodynamics or statistical physics, we do combinatorics any time we play card games.
对于那些可能不了解热力学或统计物理学领域的听众来说，我们每次玩纸牌游戏时都会进行组合数学。
You know, so you can ask yourself, with a deck of 52 cards and the four suits, how many ways are there to get three of a kind versus two of a kind or a straight?
你知道，所以你可以问问自己，对于一副 52 张牌和四种花色，有多少种方法可以得到三张同花牌、两张同花牌或顺子？
Those kinds of counting questions or probability questions, that’s part of what gave rise to combinatorics, but now it’s used all the time when we have to find clever time-saving ways to count things or to organize things, as you said, in some very structured manner.
这类计数问题或概率问题是组合数学产生的一部分，但现在当我们必须找到聪明的节省时间的方法来计算事物或组织事物时，正如您所说，以某种非常结构化的方式，它一直在使用。
Yeah, that’s a great analogy.
是的，这是一个很好的类比。
That’s exactly how I think of it.
我就是这么想的。
If you’re handed five cards, what are your possible combinatorics?
如果你有五张牌，你可能的组合是什么？
Your first card is a king, then what are the possible combinations that follow with the next four cards, and then the next three, and then the next two, and then the one?
你的第一张牌是一张国王，那么接下来的四张牌，然后是接下来的三张牌，然后是接下来的两张牌，然后是一张牌，可能的组合是什么？
Right.
正确的。
and you’ve put your finger on the key word there, right?
你已经把手指放在了那里的关键词上，对吧？
Combinatorics coming from the idea of many combinations of possibilities.
组合数学来自多种可能性组合的想法。
It feels like the beauty that we see in mathematics has a lot to do with the beauty that we see in music or in poetry in terms of elegance or the symmetry, the harmony.
感觉我们在数学中看到的美与我们在音乐或诗歌中看到的优雅、对称、和谐之美有很大关系。
There’s a strong correlation there.
那里有很强的相关性。
But one of the things that came up that I really would like to have your take on before we dive into the episode is the question of beauty.
但在我们深入讨论这一集之前，我真的很想听听你的看法，其中一件事是美的问题。
Because I know that in physics, you all use beauty occasionally as a cri- oh, okay, so you’re smirking at me here.
因为我知道，在物理学中，你们偶尔都会用美作为一种批评——哦，好吧，所以你们在这里对我傻笑。
I am!
我是！
Well, I feel like, I feel like physicists are the last ones with a straight face that really talk about beauty as a very serious criteria.
嗯，我觉得，我觉得物理学家是最后一批板着脸真正把美作为一个非常严肃的标准来谈论的人。
You know, something looks really messy, ugly, long, not compact, you suspect you’ve done something wrong or there’s something deeper worth pursuing, and that, we might dismiss it, but it seems to work.
你知道，有些东西看起来真的很乱、丑陋、冗长、不紧凑，你怀疑自己做错了什么，或者有更深层的东西值得追求，我们可能会忽视它，但它似乎有效。
That’s the interesting and sometimes uncanny thing, right?
这是有趣的，有时甚至是不可思议的事情，对吧？
That, that first of all, beauty, even just as an aesthetic issue, is hard to define, but then somehow we expect that the universe is going to have such good taste that it’s gonna…
首先，美，即使只是作为一个美学问题，也很难定义，但不知何故，我们期望宇宙会有如此好的品味，以至于它会……
You know, that we can use beauty as a selection criterion for our theories.
你知道，我们可以用美作为我们理论的选择标准。
Yeah.
是的。
And I don’t know if that is still as forceful a kind of trend, but it has been incredibly successful, and we can quantify symmetries, you know, certain organizational structures, compact expressions, grouping things together so that it seems to streamline.
我不知道这是否仍然是一种强有力的趋势，但它已经取得了令人难以置信的成功，我们可以量化对称性，你知道，某些组织结构，紧凑的表达，将事物分组在一起，以便它看起来是流线型的。
These are pretty compelling definitions of beauty, I think, in this context.
我认为，在这种情况下，这些都是对美的非常引人注目的定义。
Uh-huh.
嗯嗯。
Well, so that’s where we’re gonna go first.
好吧，这就是我们首先要去的地方。
We wanna hear Lauren Williams’s take on the issue of beauty, not in physics, but in pure math.
我们想听听劳伦·威廉姆斯对美问题的看法，不是在物理方面，而是在纯数学方面。
She is a recent MacArthur Genius Prize winner, a mathematician at Harvard University, and, uh, you’ll see in, in our conversation that she seems to specialize in finding connections between fields or topics that don’t seem like they’re related.
她是最近的麦克阿瑟天才奖获得者，哈佛大学的数学家，呃，你会在我们的谈话中看到，她似乎专注于寻找看似不相关的领域或主题之间的联系。
She finds a common thread, often not just with combinatorics, but with a very specific object in combinatorics that goes by the even more intimidating name of the positive Grassmannian.
她发现了一个共同点，通常不仅与组合学有关，而且还与组合学中一个非常具体的对象有关，该对象的名字更令人生畏，即正格拉斯曼函数。
So get ready to go plunge into those waters.
因此，准备好跳入这些水域吧。
Well, Grassmann was amazing, so I’m already interested.
嗯，格拉斯曼太棒了，所以我已经很感兴趣了。
Welcome to The Joy of Why .
欢迎来到“为什么的喜悦”。
Thank you so much.
太感谢了。
It’s great to be here.
很高兴来到这里。
The thing that’s got me really fired up to talk to you, Lauren, I have to say, besides all the beautiful math that you’ve done, I feel like your work touches on this uncanny aspect of math that has a whole philosophical dimension to it.
让我真的很兴奋地想和你交谈，劳伦，我不得不说，除了你所做的所有美丽的数学之外，我觉得你的工作触及了数学的这个不可思议的方面，它具有完整的哲学维度。
If you don’t mind, let’s start with some personal stuff.
如果你不介意的话，让我们从一些个人的事情开始吧。
I have two daughters, my wife and I, and if I understand right, you are one of four daughters?
我有两个女儿，我和妻子，如果我没理解错的话，你是四个女儿之一？
That’s right.
这是正确的。
I’m the eldest of four girls.
我是四个女孩中最大的一个。
And so, are your sisters also mathematically inclined?
那么，你的姐妹们也有数学倾向吗？
Yes.
是的。
All of my sisters liked math as well.
我所有的姐妹也都喜欢数学。
The age gaps between us are two, two, and six, so when I was growing up, the house was pretty full.
我们之间的年龄差距是两岁、两岁、六岁，所以当我长大的时候，家里已经挤满了人。
I remember teaching my youngest sister, Genevieve, addition of numbers with carries when she was probably four or five, and I showed her how to do it, and then I gave her, you know, some homework.
我记得我最小的妹妹吉纳维芙（Genevieve）大概四五岁的时候就教过带进位的数字加法，我向她展示了如何做，然后我给了她一些家庭作业。
She came back and she had a big grin on her face, and she had done it a different way.
她回来了，脸上露出灿烂的笑容，而她却以不同的方式做到了。
She was doing things left to right instead of right to left, and she thought that was hysterical.
她做事是从左到右，而不是从右到左，她认为这是歇斯底里的。
So I don’t want to put words in their mouths, but my recollection is that it seemed fun for them.
所以我不想把话放到他们嘴里，但我的记忆是，这对他们来说似乎很有趣。
And with my sister Eleanor, who’s closest to me in age, when I was in fourth or fifth grade and she was two years younger, we decided that instead of learning another language, we could learn codes, and so we started learning about Caesar cipher and then making up our own codes.
和我年龄最接近的妹妹埃莉诺在一起，当我上四年级或五年级时，她比我小两岁，我们决定不学习另一种语言，而是学习代码，所以我们开始学习凯撒密码，然后编写我们自己的代码。
We spent a lot of time doing that, and it was a lot of fun.
我们花了很多时间来做这件事，而且很有趣。
It was a secret language.
这是一种秘密语言。
I also saw some pictures of you, looking on the internet, playing the violin, and it makes me wonder if your interest in beautiful things, whether in music or in math…
我还看到了一些你在网上看、拉小提琴的照片，这让我想知道你是否对美丽的事物感兴趣，无论是音乐还是数学……
I mean, are you generally interested in beauty?
我的意思是，你一般对美感兴趣吗？
Very much so.
非常如此。
I started playing the violin when I was four, it was quite a big passion of mine.
我四岁时开始拉小提琴，这是我的一大爱好。
I was also writing poetry and stories as well.
我也写诗和故事。
But, for sure, it feels like the beauty that we see in mathematics has a lot to do with the beauty that we see in music or in poetry in terms of elegance or the symmetry, the harmony.
但是，可以肯定的是，我们在数学中看到的美与我们在音乐或诗歌中看到的优雅、对称、和谐之美有很大关系。
There’s a strong correlation there.
那里有很强的相关性。
I don’t know if learning music early on helped me to appreciate math or made me more inclined to math, or if having a mathematical appreciation also went hand in hand with having an appreciation of music.
我不知道早期学习音乐是否帮助我欣赏数学或让我更倾向于数学，或者对数学的欣赏是否也与对音乐的欣赏齐头并进。
I really don’t know.
我真的不知道。
Now, if I can fast forward a little bit can you unpack for us what is the idea of combinatorics?
现在，如果我可以快进一点，您能为我们解开组合学的概念吗？
What are the kinds of questions you would study in that field?
您在该领域会研究哪些类型的问题？
Roughly speaking, combinatorics is the study of finite or discrete structures.
粗略地说，组合学是对有限或离散结构的研究。
When one starts out learning combinatorics, one tends to learn a lot of techniques for counting or studying different kinds of objects and counting them with the usual sort of non-negative integers, zero, one, two, and so on.
当一个人开始学习组合学时，人们往往会学习很多计算或研究不同类型的对象的技术，并用通常的非负整数（零、一、二等）来计算它们。
Actually, probably 30, 40 years ago, combinatorics was looked down on as a field of math.
事实上，大概在 30、40 年前，组合数学作为一个数学领域被人看不起。
There was a viewpoint among purists that combinatorics was sort of a bag of tricks.
纯粹主义者中有一种观点认为组合学有点像一堆技巧。
But in fact, my advisor, Richard Stanley, was one of the people who really made combinatorics a much more respectable field.
但事实上，我的顾问理查德·斯坦利（Richard Stanley）是真正使组合数学成为一个更受尊敬的领域的人之一。
He wrote these beautiful books.
他写了这些美丽的书。
His work sort of pointed out that there was a lot more structure and deeper theory behind it all.
他的工作有点指出这一切背后有更多的结构和更深层次的理论。
And in particular, he has a lot of wonderful work in algebraic combinatorics, which is using techniques from algebra to study combinatorial problems.
特别是，他在代数组合学方面有很多精彩的工作，即使用代数技术来研究组合问题。
So I, I wanna go back to something you said a minute ago, the idea that some parts of math are considered more peripheral or more marginal, like they can come in and out of fashion, and I, I’m just wondering if you could expand because it gets into the sociology of math.
所以我，我想回到你刚才所说的，数学的某些部分被认为更外围或更边缘，就像它们可以流行和过时一样，我，我只是想知道你是否可以扩展，因为它进入了数学社会学。
It is a sociological phenomenon.
这是一种社会学现象。
I think partly it has to do with the age of a field.
我认为部分原因与该领域的年龄有关。
So for example, number theory is the study of numbers and their arithmetic properties, things like primes.
例如，数论是对数字及其算术性质（例如素数）的研究。
It’s a very, very old subject and had a very prestigious reputation.
这是一个非常非常古老的学科，并且享有很高的声誉。
You know, Fermat’s last theorem is one of the crowning achievements in number theory, whereas somehow combinatorics didn’t really come into its presence, good reputation, until fairly recently.
你知道，费马大定理是数论中的最高成就之一，而组合数学直到最近才真正出现并享有盛誉。
I mean, in fact, Richard Stanley was a PhD student at Harvard in the 1960s, and at the time, Harvard had no professor doing combinatorics.
我的意思是，事实上，理查德·斯坦利 (Richard Stanley) 是 20 世纪 60 年代哈佛的一名博士生，当时哈佛没有教授做组合学。
He wanted to work with Gian-Carlo Rota, who was a combinatorist working at MIT, and he had faculty members at Harvard telling him, you know, “Stop doing this Mickey Mouse mathematics.”
他想与吉安·卡洛·罗塔（Gian-Carlo Rota）合作，吉安·卡洛·罗塔是麻省理工学院的一名组合学家，哈佛大学的教员告诉他，“别再做这种米老鼠数学了。”
Fortunately, he persisted, and he, and he changed the field.
幸运的是，他坚持了下来，他，也改变了领域。
And then, you know, more recently, in the last 10 years, the mathematician June Huh won a Fields Medal largely for his work at the interface of combinatorics and algebraic geometry, and that, of course, it’s maybe a sign that this has finally been recognized.
然后，你知道，最近，在过去的 10 年里，数学家 June Huh 赢得了菲尔兹奖，主要是因为他在组合数学和代数几何接口方面的工作，当然，这可能是一个迹象，表明这一点终于得到了认可。
You’re really putting your finger on it there.
你真的把手指放在了那里。
This idea of the interconnectedness of math itself is part of a status of a subject.
数学本身的相互关联性的想法是学科地位的一部分。
If a subject is seen as too far in the outskirts, not connecting to the other domains in math, that tends to lower its reputation.
如果一门学科被认为过于偏远，与数学的其他领域没有联系，那么它的声誉往往会降低。
If we have complete freedom to ask whatever question we want, well, what I want to do is find questions with beautiful answers.
如果我们有完全的自由来问我们想要的任何问题，那么，我想做的就是找到带有漂亮答案的问题。
Right?
正确的？
Whereas the things that are in the core, whatever that means, have higher status like algebraic geometry you mentioned.
而核心的东西，无论这意味着什么，都有更高的地位，比如你提到的代数几何。
So I began by asking you questions about beauty, the aesthetic dimension of math, and so I would like to try to connect that to a quote that I read of yours.
所以我首先问你关于美、数学的美学维度的问题，所以我想尝试将其与我读到的你的一句话联系起来。
It says, “If you ask a question and the answer is not beautiful, that means you asked the wrong question.” I don’t know.
它说：“如果你提出一个问题而答案并不漂亮，那就意味着你问错了问题。”我不知道。
I’d like to hear more.
我想听更多。
Yes.
是的。
So starting from the beginning of my time as a graduate student, I was going to seminars, I was taking classes, and one of the things I was trying to figure out was, what do different people find interesting in mathematics, or what do they find beautiful in mathematics?
因此，从我作为研究生开始，我就去参加研讨会，上课，我试图弄清楚的一件事是，不同的人觉得数学中的什么有趣，或者他们觉得数学中的什么美丽？
What do I find interesting?
我觉得什么有趣？
What do I find beautiful?
我觉得什么好看？
And I picked up this aesthetic in part from my advisor.
我的这种审美观部分是从我的导师那里学到的。
You know, the work should be beautiful.
你知道，工作应该是美丽的。
Uh.
呃。
And as a pure mathematician, we’re not beholden to particular questions or applications, and so what are the standards to which we should hold ourselves?
作为一名纯粹的数学家，我们不受制于特定的问题或应用，那么我们应该坚持什么标准呢？
Well, we should be looking for something beautiful.
好吧，我们应该寻找美丽的东西。
And we have complete freedom as pure mathematicians, and if we have complete freedom to ask whatever question we want, well, what I want to do is find questions with beautiful answers.
作为纯粹的数学家，我们拥有完全的自由，如果我们有完全的自由来提出我们想要的任何问题，那么我想做的就是找到具有美丽答案的问题。
And I think those are the questions and answers that tend to wind up becoming the most important or having the most connections to other fields, whether it’s within math or outside of math.
我认为这些问题和答案往往会成为最重要的或与其他领域最有联系的问题和答案，无论是在数学内部还是数学之外。
In the popular mind, math is known as being a very black and white subject.
在大众心目中，数学被认为是一门非黑即白的学科。
There’s a right answer, there’s a wrong answer.
有正确答案，也有错误答案。
When you start talking about beauty, it sounds very subjective.
当你开始谈论美时，听起来很主观。
There’s the old line about beauty being in the eye of the beholder.
有句老话说情人眼里出西施。
And it seems a little surprising that mathematicians would be so obsessed with beauty, given that it’s so hard to define what beauty means.
数学家如此痴迷于美似乎有点令人惊讶，因为很难定义美的含义。
And I guess I’d like to know first of all, what would it look like for math to be not beautiful?
我想我首先想知道，数学不美丽会是什么样子？
So a beautiful piece of math is short, clean, elegant, maybe a statement that connects two mathematical objects that you didn’t expect to be connected.
因此，一段优美的数学是简短、干净、优雅的，也许是一个连接两个你意想不到的数学对象的陈述。
That is something that I would find more likely to be beautiful than if the theorem takes paragraphs or pages to even get to the statement.
我发现这比定理需要段落或页面才能到达陈述更可能是美丽的。
On the other hand, one could come up with a true theorem or a true fact that seems very arbitrary or specific.
另一方面，人们可以提出一个看起来非常武断或具体的真实定理或真实事实。
Maybe you ask like, “Okay, what if I take 73 random lines in the plane, and then I look at the, I don’t know, I look at some curve going through these four intersection points, what’s gonna happen?
也许你会问，“好吧，如果我在平面上随机取 73 条线，然后我看，我不知道，我看一些穿过这四个交点的曲线，会发生什么？
You know, that’s less likely to be considered beautiful than a statement like there are infinitely many primes.
你知道，与素数有无穷多个这样的说法相比，这不太可能被认为是美丽的。
You know, a statement that is short and simple, clean, and feels more sort-of universal is going to be considered, considered more beautiful than the other kind of statement.
你知道，一个简短、简单、干净、感觉更通用的声明将会被考虑，被认为比其他类型的声明更漂亮。
So let me see if I follow you.
所以让我看看我是否关注你。
The aesthetic you’re talking about tends to like economy of means or minimalism.
你所说的审美往往喜欢经济手段或极简主义。
You like universality.
你喜欢普遍性。
You don’t want a lot of very highly specific things.
你不需要很多非常具体的东西。
To the extent that it’s general, that’s a plus.
就其通用性而言，这是一个优点。
To the extent that it’s compact, that helps.
就其紧凑而言，这很有帮助。
But I was hearing some disrespect towards certain types of numbers in favor of other types of numbers.
但我听到一些人对某些类型的数字不尊重，而偏爱其他类型的数字。
Well, I am a combinatorialist after all.
好吧，我毕竟是一个组合主义者。
But, but it was more about questions where the answer is a theorem that holds true for infinitely many cases, as opposed to a question with a single answer, which is some real specific number.
但是，它更多的是关于答案是一个适用于无限多种情况的定理的问题，而不是只有一个答案的问题，即一些真正的具体数字。
So in terms of, um, like if we were doing a self-help manual for mathematicians, and you suppose you’re studying something and it’s coming out not beautiful according to these and maybe some other criteria, what do you do?
所以，嗯，就像我们正在为数学家编写一本自助手册，你假设你正在研究一些东西，但根据这些标准和其他一些标准，结果并不漂亮，你会做什么？
Like, are there moves that you can make to push yourself on a more beautiful track?
比如，你可以采取哪些措施来推动自己走上更美丽的轨道？
I guess I would try to tweak the question.
我想我会尝试调整这个问题。
If it seems like I’m arriving at the answer, but it’s simply not elegant, doesn’t seem to be interesting, then I would try to tweak the question.
如果我似乎找到了答案，但它根本不优雅，似乎不有趣，那么我会尝试调整问题。
I mean, I might in an extreme situation just abandon that effort, but if I feel like there should be something interesting nearby, then I would try to adjust the question.
我的意思是，在极端情况下我可能会放弃这种努力，但如果我觉得附近应该有一些有趣的东西，那么我会尝试调整问题。
And that’s the nice gambit because in a lot of fields you’re sort of stuck with the question.
这是一个很好的策略，因为在很多领域你都被这个问题困住了。
But in math, you mentioned the freedom to think about whatever they want, and so you’re saying you can leverage that freedom to maybe change the question.
但在数学中，你提到了思考他们想要的任何东西的自由，所以你说你可以利用这种自由来改变问题。
Maybe your ugly answer is a sign that your question isn’t quite right.
也许你丑陋的答案表明你的问题不太正确。
Okay.
好的。
So now I think I understand what you meant by that.
所以现在我想我明白你的意思了。
Yeah.
是的。
I would imagine that that people in music or art could relate to this kind of statement.
我想音乐或艺术领域的人可能会认同这种说法。
Maybe there’s a theme that they’re trying to play with in composing a piece of music and it doesn’t have the right harmonies.
也许他们在创作一首音乐时试图演奏一个主题，但它没有正确的和声。
I’m sure there’s a lot of tweaking of the framework or tweaking of the approach that goes into art that results in a better end product.
我确信对艺术的框架或方法进行了大量的调整，从而产生更好的最终产品。
Well, so I wanna dive into that now, this set of ideas related to a thing called the Grassmannian and the positive Grassmannian, which for a long time were a playground for pure mathematicians.
好吧，所以我现在想深入探讨这一点，这组想法与一种叫做格拉斯曼和正格拉斯曼的东西有关，在很长一段时间内，它们是纯数学家的游乐场。
So okay, what the heck is this thing?
那么好吧，这到底是什么东西？
For people who have never heard of the Grassmannian, start us off.
对于那些从未听说过格拉斯曼尼亚的人，请让我们开始吧。
How should we think about it in the simplest cases?
在最简单的情况下我们应该如何思考？
Yeah.
是的。
So the Grassmannian, it’s a geometric object, and it’s a bit like a library for keeping track of simpler objects.
所以格拉斯曼，它是一个几何对象，它有点像一个用于跟踪更简单对象的库。
So, for example, the Grassmannian of one-planes in two-space is the set of all lines through the origin in the x-y plane.
例如，二空间中一平面的格拉斯曼方程是 x-y 平面中通过原点的所有直线的集合。
In the Grassmannian, each line passes through the northern half of the sphere.
在格拉斯曼方程中，每条线都穿过球体的北半部。
Samuel Velasco/Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
And so we’re, we’re thinking of this as a sort-of one mathematical object that’s keeping track of all of these lines.
所以我们认为这是一种跟踪所有这些线的数学对象。
And so you can imagine all these lines passing through the origin in two-dimensional space, and then the positive part of this Grassmannian would be the set of those lines that go through the positive orthant.
所以你可以想象所有这些线都穿过二维空间中的原点，然后这个格拉斯曼函数的正部分将是那些穿过正原点的线的集合。
Let me think about these lines.
让我思考一下这些线条。
If I were looking at a compass, there’s a line that goes north and south.
如果我看指南针，就会看到一条向北和向南的线。
I think of that as one line.
我认为这是一条线。
So north and south are both part of the same line.
所以南北都是同一条线的一部分。
And east and west are also part of the same line.
而且东边和西边也是在同一条线上的。
That’s right.
没错。
And then there’s sort of infinitely many lines that we can get by tilting either of those lines.
然后，通过倾斜其中任何一条线，我们可以得到无限多条线。
Yeah.
是的。
So I’m thinking about infinitely many different lines, and you’re telling me there’s a way I could think about them as somehow described by a single shape?
所以我正在考虑无限多种不同的线条，而你告诉我有一种方法可以将它们视为由单一形状来描述？
Yes.
是的。
And if you like, if you’re thinking about this set of all lines through the origin in two-dimensional space, each one passes through the sort-of northern half of your compass.
如果你愿意的话，如果你在二维空间中思考这组穿过原点的所有线，每条线都会穿过指南针的北半部。
And so you could actually just keep track of each of those lines by the intersection point with a circle.
因此，您实际上可以通过与圆的交点来跟踪每条线。
And then if you just think about this positive semicircle at the top, then each point on that sort-of positive semicircle will be specifying uniquely a line that goes through the origin.
然后，如果您只考虑顶部的正半圆，那么该正半圆上的每个点都将唯一指定一条穿过原点的线。
Which on a compass would just have little markings on it that say north, northeast, we could even have north-north-north-northeast.
在指南针上只有很少的标记，上面写着北、东北，我们甚至可以有北-北-北-东北。
I mean, that gets cumbersome to do it that way.
我的意思是，这样做会很麻烦。
But you’re saying there’s infinitely many points on this northern half circle, and then the totality of all those points makes what you called a semicircle.
但是你说这个北半圆上有无穷多个点，然后所有这些点的总和就形成了你所谓的半圆。
But do I consider…
但我是否考虑...
What’s going on with the equator or the east and western points?
赤道或东西方点是怎么回事？
That’s, that’s right.
就是这样，没错。
We should only include one of them because there’s only one …
我们应该只包含其中之一，因为只有一个......
there’s one line that passes through both the west and the east-most points.
有一条线穿过最西点和最东点。
Yeah.
是的。
So we should only keep one of them and throw the other point away.
所以我们应该只保留其中的一点，而丢弃另一点。
As far as the collection of all these lines, am I right in thinking that it’s like a line segment that goes from east, but I don’t include east, all the way around the top to west, but I do include west?
就所有这些线的集合而言，我是否认为它是从东出发的线段，但我不包括东，一直绕到顶部到西，但我确实包括西？
So is it like a line segment with one end point but not the other endpoint?
那么它是不是像一条有一个端点但没有另一个端点的线段呢？
Exactly, yes.
没错，是的。
That would be the Grassmannian of 1-planes in 2-space.
这就是 2 空间中 1 平面的格拉斯曼方程。
We, that’s how we can think about it.
我们，这就是我们可以考虑的方式。
You could think of it as being like a half-open interval.
您可以将其视为半开区间。
Huh.
呵呵。
Okay.
好的。
That’s a kind of a weird object.
这是一种奇怪的物体。
I mean, it feels kind of unsymmetrical, a half-open interval, like I’m, I’m dying to close that other endpoint, but I shouldn’t.
我的意思是，感觉有点不对称，半开区间，就像我一样，我很想关闭另一个端点，但我不应该。
Not if I want the Grassmannian.
如果我想要格拉斯曼式的就不行。
Well, we sort of identify those two points, so it’s almost like we could think about sort of gluing those together.
好吧，我们已经确定了这两点，所以我们几乎可以考虑将它们粘合在一起。
Okay.
好的。
And then for the positive Grassmannian, you only want the positive orthant.
然后对于正的格拉斯曼，你只想要正的orthant。
Right.
对。
So first the Grassmannian of lines in three-dimensional space in this slightly higher dimensional setting would look more or less like the northern hemisphere.
因此，首先，在这个稍高维度的环境中，三维空间中的格拉斯曼线看起来或多或少像北半球。
And then if we restrict to the positive part of the Grassmannian, then we’re just looking at the intersection of that hemisphere with the positive orthant.
然后，如果我们限制格拉斯曼函数的正部分，那么我们只需考虑该半球与正正交部分的交点。
And so it’s actually gonna look like a triangle, like a kind of a curvy triangle.
所以它实际上看起来像一个三角形，就像一种弯曲的三角形。
The positive Grassmannian is the part of the Grassmannian in which all the lines have non-negative slope.
正格拉斯曼阶是格拉斯曼阶中所有线都具有非负斜率的部分。
Samuel Velasco/Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
Okay.
好的。
And this word “orthant,” which isn’t totally familiar, is the 3D version of quadrant.
“orthant”这个词大家不太熟悉，它是象限的 3D 版本。
Right.
对。
There’s eight of them.
有八个。
So that’s why you’re saying “orthant.”
这就是为什么你说“orthant”。
Yeah.
是的。
So that positive orthant would be where the X, Y, and Z coordinates are all positive or non-negative.
因此，正坐标就是 X、Y 和 Z 坐标均为正或非负的位置。
Okay.
好的。
And you say it looks sort of like a curvy triangle.
你说它看起来有点像一个弯曲的三角形。
And so at this point, if people are still with us, why would anyone think about this object?
所以在这一点上，如果人们仍然和我们在一起，为什么会有人想到这个物体呢？
This doesn’t seem like an obvious thing to think about.
这似乎不是一件显而易见的事情。
Yeah, yeah.
是啊是啊。
It wasn’t an obvious thing to think about, but back in the 1900s, mathematicians were studying certain kinds of matrices called totally positive matrices, and they had nice properties.
这并不是一件显而易见的事情，但早在 1900 年代，数学家们就在研究某些称为全正矩阵的矩阵，它们具有很好的性质。
They had some connections to different systems like oscillation.
它们与不同的系统有一些联系，比如振荡。
And then in the late 1990s, early 2000s, Lusztig and Postnikov realized that there was a way to sort of generalize this notion of totally positive matrices to an object that lived inside the Grassmannian.
然后在 20 世纪 90 年代末、2000 年代初，Lusztig 和 Postnikov 意识到有一种方法可以将完全正矩阵的概念推广到格拉斯曼内部的对象。
And so it was just sort of a purely interesting mathematical idea to try to study total positivity, not just for matrices anymore, but for geometric objects like the Grassmannian.
因此，尝试研究完全正性只是一种纯粹有趣的数学想法，不仅适用于矩阵，而且适用于像格拉斯曼这样的几何对象。
So, there’s all kinds of sets of matrices that describe motions and symmetries for the real world.
因此，有各种各样的矩阵组来描述现实世界的运动和对称性。
And they come up in quantum theory.
它们出现在量子理论中。
They’re used all the time now in artificial intelligence, but even inside of math, as you say, they’re, they can act like machines that do things to other mathematical objects.
它们现在一直在人工智能中使用，但即使在数学内部，正如你所说，它们也可以像机器一样对其他数学对象做事。
Yes, in math, one of the common themes is that we study not just the mathematical objects, but also the relationships between the objects.
是的，在数学中，共同的主题之一是我们不仅研究数学对象，还研究对象之间的关系。
And matrices can give us a way to create or to analyze relationships between different mathematical objects.
矩阵可以为我们提供一种创建或分析不同数学对象之间关系的方法。
Now, one of the results that you are known for was this positive Grassmannian that we talked about, you looked at in a combinatorial way, in the very general case.
现在，您所熟知的结果之一是我们谈到的正格拉斯曼函数，您在非常一般的情况下以组合方式进行观察。
So, tell us a little bit of the flavor of what you did there.
那么，请告诉我们一些您在那里所做的事情的味道。
Yeah, absolutely.
是的，绝对是。
Back when I was a grad student or postdoc, I think I had a conversation with some other mathematicians about, you know, just what combinatorics is as a field.
当我还是一名研究生或博士后时，我想我与其他一些数学家进行了一次对话，内容是关于组合数学作为一个领域是什么。
and one thing that we discussed at that dinner was that one can think of combinatorics, not necessarily just as a field, but as an attitude.
我们在那次晚宴上讨论的一件事是，人们可以将组合学视为一种态度，而不必只是一个领域。
You know, we can go through life, uh, with a combinatorial attitude and take a combinatorial approach to different problems.
你知道，我们可以以组合的态度来度过生活，并对不同的问题采取组合的方法。
And the positive Grassmannian can be divided into pieces of different dimensions.
而正格拉斯曼则可以分为不同维度的碎片。
An analogy I like to use is that of the cube, say the three-dimensional cube.
我喜欢用立方体做类比，比如三维立方体。
If a combinatorialist looks at it, they may come away saying, “Well, it has six two-dimensional faces,” these squares on the different sides, “and it has 12 one-dimensional edges, and it also has eight zero-dimensional pieces,” the eight vertices.
如果组合主义者看到它，他们可能会说，“嗯，它有六个二维面”，这些正方形位于不同的边上，“它有 12 个一维边，它还有八个零维部分”，即八个顶点。
And so you can associate these numbers, six, 12, and eight to a cube.
因此，您可以将这些数字 6、12 和 8 关联到一个立方体。
That’s what a combinatorialist might do.
这就是组合主义者可能会做的事情。
Now, there are infinitely many positive Grassmanians, and they can have arbitrarily high dimension, but the first problem that I worked on in graduate school was coming up with an explicit formula for how many pieces there are of each dimension.
现在，有无限多个正的格拉斯曼函数，它们可以具有任意高的维度，但我在研究生院研究的第一个问题是提出一个明确的公式来计算每个维度有多少个部分。
So, I wrote down a polynomial that for any k and n tells you how many pieces there are of each dimension in that positive Grassmannian.
因此，我写下了一个多项式，对于任意 k 和 n，该多项式可以告诉您该正格拉斯曼函数中每个维度有多少个部分。
Okay, so let’s now make a little swerve from this pretty abstract realm of matrices and Grassmannians and positive Grassmannians to the much more mundane world of traffic and waves on the ocean and proteins being made inside of cells, because it turns out all those things can be viewed as part of one story.
好吧，现在让我们从这个相当抽象的矩阵、格拉斯曼尼亚和正格拉斯曼尼亚的领域转向更平凡的世界，海洋上的交通和波浪以及细胞内产生的蛋白质，因为事实证明所有这些东西都可以被视为一个故事的一部分。
I was floored when I saw this paper to think that my polynomials had to do with a sort of, quote-unquote, ‘real world.’
当我看到这篇论文时，我感到震惊，因为我认为我的多项式与某种“现实世界”有关。
That’s right.
没错。
There have been sort of three different areas that I’ve had personal experience with where the positive Grassmannian got connected.
我个人经历过与积极的格拉斯曼主义者有联系的三个不同领域。
During my postdoc, I learned that another mathematician, Sylvie Corteel, had written a paper which said that my polynomials that were counting pieces of the positive Grassmannian according to dimension were also computing probabilities in a model that had been introduced to study translation in protein synthesis and was also used as a model for traffic flow.
在我的博士后期间，我了解到另一位数学家西尔维·科蒂尔（Sylvie Corteel）写了一篇论文，其中说我的多项式根据维度计算正格拉斯曼函数的部分，也在计算模型中的概率，该模型已被引入研究蛋白质合成中的翻译，也被用作交通流模型。
I was floored when I saw this paper to think that my polynomials had to do with a sort of, quote-unquote, “real world.” Her result was quite beautiful.
当我看到这篇论文时，我感到很震惊，因为我认为我的多项式与某种“真实世界”有关。她的结果非常漂亮。
Basically, she was saying that my polynomials were giving the probability that in a lattice with n sites or in a road with space for n cars, there are exactly k cars present.
基本上，她是说我的多项式给出了这样的概率：在具有 n 个站点的格子中或在可容纳 n 辆汽车的道路中，恰好有 k 辆汽车存在。
That’s what my polynomials were computing.
这就是我的多项式的计算内容。
So great, so we know the probability that in a road with space for n cars, there’s exactly k present.
太好了，所以我们知道在一条可容纳 n 辆汽车的道路上，恰好有 k 辆汽车存在的概率。
Well, what if we want to know the probability that the cars are present in positions one, four, five, eight?
那么，如果我们想知道汽车出现在位置一、四、五、八的概率怎么办？
You know, what if you want to know all the probabilities that any given configuration of cars is there?
你知道，如果你想知道任何给定的汽车配置存在的所有概率怎么办？
And so that was the natural question to ask.
所以这是很自然要问的问题。
That actually kicked off a decades-long collaboration with Sylvie.
这实际上拉开了与西尔维长达数十年的合作的序幕。
I mean, we’ve written a number of papers together by now.
我的意思是，到目前为止我们已经一起写了很多论文。
[ STROGATZ: So, there’s a lot to unpack there, Janna.
[STROGATZ：所以，那里有很多东西需要解开，Janna。
Did that wash over you?
这有洗掉你吗？
Well, I mean, I grasp some of it, right?
嗯，我的意思是，我掌握了一些，对吧？
This idea that you can cut some mathematical object into pieces, find some mathematical rule for the number of pieces occurring of a certain variety.
这个想法是，你可以将一些数学对象切成碎片，找到某种数学规则来计算某种类型的碎片的数量。
She was saying dimensionality.
她说的是维度。
So having heard all of that, which is very intriguing, just give me the bird’s-eye view of what a Grassmannian is.
听完所有这些非常有趣的内容后，请让我鸟瞰一下什么是格拉斯曼主义者。
Okay, fair enough.
好吧，很公平。
Right, it’s not a concept we run into every day.
是的，这不是我们每天都会遇到的概念。
So here’s what it is.
这就是它是什么。
There’s a technical way we could define it, but before I give you that, can I give you ‘What’s it gonna do for us?’ How is it helpful?
我们可以通过一种技术方式来定义它，但在我给你这个之前，我可以告诉你“它会对我们做什么？”它有什么帮助？
So, it’s a really nice meta-concept.
所以，这是一个非常好的元概念。
It’s a shape that tells us about other shapes.
这个形状可以告诉我们其他形状。
It’s a shape that can be used as a library or a catalog for other kinds of structures or shapes.
它是一种可以用作其他类型结构或形状的库或目录的形状。
So it’s one shape or it’s a family of shapes?
那么它是一种形状还是一系列形状？
It’s a family of shapes.
这是一个形状家族。
There’s different Grassmannians.
有不同的格拉斯曼主义者。
I mean, here’s the technical definition, which may work for you, but I don’t wanna linger on it too long because I don’t think it’s the most helpful way to think about it.
我的意思是，这是技术定义，可能对您有用，但我不想在它上面停留太久，因为我认为这不是思考它的最有帮助的方式。
Technically, it has to do with thinking about all the different ways that k-dimensional spaces, linear spaces, like a two-dimensional space would be a plane, a one-dimensional space would be a line, a three-dimensional space is what we’re used to for ordinary 3D space.
从技术上讲，它与思考 k 维空间、线性空间（例如二维空间是平面、一维空间是直线、三维空间是我们习惯的普通 3D 空间）的所有不同方式有关。
Yeah.
是的。
So you’re trying to think about the totality of all k-dimensional linear spaces through the origin of n-dimensional space.
所以你试图通过 n 维空间的起源来思考所有 k 维线性空间的总体。
So there’s two parameters, k and n.
所以有两个参数，k 和 n。
Now, the simplest case would be think about lines through the origin.
现在，最简单的情况是考虑通过原点的线。
That’d be one-dimensional spaces through the origin in a plane.
那将是通过平面原点的一维空间。
Right.
正确的。
That would be the 1-2 Grassmannian or something like that.
那将是 1-2 格拉斯曼或类似的东西。
Okay?
好的？
So there’s, infinitely many, a whole continuum of lines, but if you wanted to parameterize them in our language, if you wanted to catalog them, you could do it by saying, “What’s their compass direction?” Like there’s the line that goes north-south, or there’s the line that goes north-northeast and south-southwest or something like that, right?
因此，有无限多的连续线，但如果你想用我们的语言对它们进行参数化，如果你想对它们进行编目，你可以通过说“它们的罗盘方向是什么？”来做到这一点。就像有一条南北走向的线，或者有一条东北北和西南西南的线，对吧？
So if I listed all of those possible lines, it could be the whole upper semicircle.
因此，如果我列出所有这些可能的线，它可能是整个上半圆。
So that’s a shape.
这就是一个形状。
And so people study different Grassmannians, some high-dimensional space and some lower dimensional.
因此人们研究不同的格拉斯曼空间，一些是高维空间，一些是低维空间。
Exactly.
确切地。
Now, Lauren specializes in this piece of it that’s called the positive Grassmannian, which in our little example with lines through the origin in the plane would be like only considering the ones that have positive slope.
现在，劳伦专门研究这一部分，称为正格拉斯曼曲线，在我们的小例子中，通过平面原点的线就像只考虑具有正斜率的线一样。
And that turns out to have extra structure that makes it more helpful in lots of applications.
事实证明，它具有额外的结构，使其在许多应用程序中更加有用。
At this point, it seems like something that pure geometers would think about.
在这一点上，这似乎是纯几何学家会思考的问题。
This is about a shape that classifies other shapes.
这是关于对其他形状进行分类的形状。
The spooky thing is that this pops up all over the place in real-world settings.
令人毛骨悚然的是，这种情况在现实世界中随处可见。
So like she mentions, traffic flow.
就像她提到的，交通流量。
I want you to have not an image of cars motoring down the highway, ’cause she doesn’t really mean that kind of traffic.
我希望你不要想象汽车在高速公路上行驶，因为她并不是真正指那种交通。
Think of back when COVID was rampant and we had to stand in line at the checkout for the supermarket, and you had to stay six feet behind the person in front of you, right?
回想一下，当新冠疫情肆虐时，我们必须在超市收银台排队，而你必须与前面的人保持六英尺的距离，对吗？
So imagine you had something like 10 spots available that you could stand on.
想象一下，您有大约 10 个可以站立的位置。
That would be like our n.
那就像我们的n。
And now people start arriving to get in line and also people at the front of the line can leave, and the rules of the game are that whatever spot you’re on, you have some probability of moving forward one spot, except not if someone’s standing there.
现在人们开始排队，排在前面的人也可以离开，游戏规则是，无论你在哪里，你都有一定的概率向前移动一个位置，除非有人站在那里。
There’s a constraint.
有一个限制。
If you let this whole thing run for a long time with people arriving at random and leaving at random, and moving forward one spot at random when they can, you could classify all the possible ways that these 10 spots could be occupied by four people, let’s say.
如果你让整个过程持续很长时间，人们随机到达、随机离开，并在可能的情况下随机向前移动一个位置，那么你可以对这 10 个位置可能被四个人占用的所有可能方式进行分类。
That would be the k.
那就是 k。
It turns out the 4,10 Grassmannian tells me something about the likelihood of seeing a particular number of people in this queue.
事实证明，4,10 格拉斯曼人告诉我一些关于在这个队列中看到特定数量的人的可能性的信息。
Now, I’m curious.
现在，我很好奇。
I can imagine during COVID, as you said, having to solve this problem, right?
我可以想象在新冠疫情期间，正如你所说，必须解决这个问题，对吗？
It’s a problem that has to be solved because we now have distribution centers for vaccines, and this is happening or something like that.
这是一个必须解决的问题，因为我们现在有疫苗配送中心，而这种情况正在发生或类似的事情。
How does somebody notice that the polynomial that they’ve generated to answer this practical question happens to be the same as a polynomial a very abstract mathematician has found for a Grassmannian on the positive with positive Grass…
有人如何注意到他们为回答这个实际问题而生成的多项式恰好与一位非常抽象的数学家为正格拉斯曼方程找到的多项式相同......
I mean, how do they even notice this correlation?
我的意思是，他们是如何注意到这种相关性的？
That might be the unique genius of Lauren Williams and her collaborator, Sylvie Corteel.
这可能是劳伦·威廉姆斯和她的合作者西尔维·科蒂尔的独特天才。
And it’s not just about the queues.
这不仅仅是队列的问题。
If you think about ribosomes moving down an mRNA molecule as they’re doing protein synthesis, it also pops up in that setting.
如果你想到核糖体在进行蛋白质合成时沿着 mRNA 分子向下移动，它也会在这种情况下出现。
You see what I’m getting at?
你明白我的意思了吗？
This is a really fun, diverse set of applications all mysteriously falling under the heading of the positive Grassmannian.
这是一组非常有趣、多样化的应用程序，所有这些应用程序都神秘地属于积极的格拉斯曼式的标题下。
But after the break, Lauren Williams will walk us through why this phenomenon might be happening, why it’s happening so pervasively, and also how artificial intelligence may or may not take over mathematics.
但休息结束后，劳伦·威廉姆斯将向我们介绍为什么这种现象可能会发生、为什么它如此普遍地发生，以及人工智能可能会或可能不会接管数学。
Welcome back to The Joy of Why .
欢迎回到“为什么的喜悦”。
We’re speaking with Harvard mathematician Lauren Williams about algebraic combinatorics and the positive Grassmannian.
我们正在与哈佛大学数学家劳伦·威廉姆斯讨论代数组合学和正格拉斯曼函数。
I would love to ask about why these connections to the much more mundane world happen.
我很想问为什么会发生这些与更平凡的世界的联系。
I know that no one knows.
我知道没有人知道。
Well, you know, what’s really interesting to me is that you know, with the model of traffic flow, it’s this model of particles that repel each other.
嗯，你知道，对我来说真正有趣的是，在交通流模型中，正是这种相互排斥的粒子模型。
And with the shallow water waves, these are waves that are sort of coming together and interacting.
对于浅水波浪，这些波浪有点聚集在一起并相互作用。
And with the scattering aptitudes, it’s about particles that are being thrown together and interacting.
散射能力是指粒子被抛在一起并相互作用。
And somehow it’s always about particles or waves that are being flung together and then they sort of repel in some way.
不知何故，它总是关于粒子或波被抛在一起，然后它们以某种方式排斥。
And you know, the coordinates one uses for the Grassmannian are Plücker coordinates.
您知道，用于格拉斯曼坐标的坐标是普吕克坐标。
And if you have your k by n matrix, you think of this as a, as a list of column vectors.
如果您有 k × n 矩阵，您可以将其视为 a，作为列向量列表。
Well, if two, two vectors get so close that they’re actually on top of each other, your Plücker coordinate vanishes.
好吧，如果两个、两个向量非常接近，以至于它们实际上彼此重叠，那么您的普吕克坐标就会消失。
So there’s something in the nature of the Plücker coordinates on the Grassmannian that build in this repelling property.
因此，格拉斯曼坐标上的普吕克坐标本质上存在某种东西，建立在这种排斥性的基础上。
And so I’ve always wondered if it could be possible to connect these three different settings, whether it’s the particles repelling each other, or the waves, or the scattering aptitudes.
所以我一直想知道是否有可能将这三种不同的设置联系起来，无论是相互排斥的粒子、波还是散射能力。
But somehow I think it all comes down to Plücker coordinates on the Grassmannian.
但不知怎的，我认为这一切都归结于格拉斯曼尼安上的普吕克坐标。
The Grassmannian is so universal, and then there’s something about positivity that just captures properties of the real world for some reason.
格拉斯曼主义是如此普遍，而且出于某种原因，积极性只是捕捉了现实世界的属性。
That’s nice.
那很好。
That’s a very, very nice answer.
这是一个非常非常好的答案。
And I mean, it feels like a more refined way of saying what you said at first.
我的意思是，这感觉像是一种更精致的方式来表达你最初所说的话。
’Cause I’ve seen diagrams like when you look at the particles in the Feynman diagrams, veering towards each other and then bouncing off.
因为我见过像费曼图中的粒子一样的图表，它们相互转向然后弹开。
Or if you look at the diagrams of the water waves where you’re just looking at, I don’t know what, the crests or something, there’s ways of drawing the pictures that it almost looks like you’re drawing the same picture over and over.
或者，如果你看水波的图表，你只是在看，我不知道什么，波峰或其他东西，有一些绘制图片的方法，看起来几乎就像你一遍又一遍地绘制同一张图片。
Right, right, right.
对，对，对。
Yeah.
是的。
So, I mean, it would be weird, then again, maybe not so surprising that if at a very deep level we’re drawing this same picture over and over and nature is interpreting it, or math is interpreting it in different ways in different settings, but it’s kind of the same mechanism.
所以，我的意思是，如果在一个非常深的层次上，我们一遍又一遍地绘制同一张图，而大自然正在解释它，或者数学在不同的环境中以不同的方式解释它，那么这会很奇怪，也许并不那么令人惊讶，但这是相同的机制。
But your, your thing with the Plücker coordinates, and the…
但是你的，你的普吕克坐标，还有……
does the zero mean that that’s the analog of repulsion.
零是否意味着这与排斥力类似。
They won’t go through each other because of that zero?
他们不会因为那个零而互相穿越？
Well, they could, but then there’s a sign change.
好吧，他们可以，但随后符号发生了变化。
And then somehow, like with the shallow water wave stuff, I was thinking about why positivity comes into the picture.
然后不知何故，就像浅水波浪的东西一样，我在思考为什么积极性会出现在画面中。
You know, to analyze these solutions, to analyze these water waves, you use Soliton solutions to the KP equation, and that involves a tau function in which you take the log of a certain function, and this function is built out of the Plücker coordinates in some way.
你知道，为了分析这些解，为了分析这些水波，你需要使用 KP 方程的孤子解，这涉及到一个 tau 函数，在这个函数中你取某个函数的对数，而这个函数是以某种方式根据普吕克坐标构建的。
And as long as the Plücker coordinates are all non-negative, you’re taking the log of something that’s always positive.
只要普吕克坐标都是非负的，你所得到的对数就总是正数。
But if you lose this positivity, if you now are talking about all points in the Grassmannian and not just the positive part, you might at some point be taking the log of zero or something really close to zero.
但是，如果你失去了这种积极性，如果你现在谈论的是格拉斯曼函数中的所有点而不仅仅是正部分，那么你可能会在某个时刻取对数为零或非常接近于零的值。
But then what happens is that your model for shallow water waves goes off to plus or minus infinity, which obviously does not represent the real world.
但随后发生的情况是，您的浅水波模型会变得正负无穷大，这显然并不代表现实世界。
And so there’s something about, you know, if you want to stay in the real world, you have to stay away from this zero.
所以，如果你想留在现实世界，你就必须远离这个零。
And it means restricting to the positive Grassmannian.
这意味着仅限于正格拉斯曼。
So, yeah.
所以，是的。
But if we were to just get a little sloppier, but I think maybe more understandable about it, is it that there are sort of a bank of possible patterns that can happen in our minds or in nature.
但如果我们稍微草率一点，但我认为也许更容易理解，那就是我们的头脑或自然界中可能会发生一系列可能的模式。
And sometimes those patterns just, you know, if they’re fundamental enough, they will show up in many parts of our thought and in our observations.
有时这些模式，你知道，如果它们足够基本，它们就会出现在我们思想和观察的许多部分中。
So like there’s a certain family of patterns that you are swirling around and this positive Grassmannian story is encoding them, and they have different manifestations in math and in the world, but it’s kind of the same pattern over and over.
就像你周围有一个特定的模式家族，这个积极的格拉斯曼故事正在对它们进行编码，它们在数学和世界上有不同的表现，但它是一遍又一遍地相同的模式。
Yeah, maybe that’s right.
是的，也许是这样。
Maybe that’s right.
也许这是对的。
I mean, the Grassmannian is so universal, and then there’s something about positivity that just captures properties of the real world for some reason.
我的意思是，格拉斯曼主义是如此普遍，而且出于某种原因，积极性只是捕捉了现实世界的属性。
This story is not over because then somehow you get entangled, naturally I use that word, with things happening in quantum physics specifically with things related to a very beautiful quantum field theory: N=4 supersymmetric Yang-Mills theory.
这个故事还没有结束，因为不知何故你会陷入困境，自然地我用这个词，与量子物理学中发生的事情特别是与一个非常美丽的量子场论相关的事情：N=4超对称杨米尔斯理论。
If I’ve got that right.
如果我没猜错的话。
But anyway, Nima Arkani-Hamed and other collaborators are looking at this fantastic model and somehow you connect to them.
但无论如何，Nima Arkani-Hamed 和其他合作者正在研究这个奇妙的模型，并且您以某种方式与他们建立了联系。
You want to build that bridge for us?
你想为我们建造那座桥吗？
Yes.
是的。
So they started to realize that somehow the structure of the positive Grassmannian was helping to understand scattering amplitudes.
因此，他们开始意识到，正格拉斯曼结构在某种程度上有助于理解散射振幅。
So scattering amplitudes are basically probabilities that tell you what you might expect would happen if you throw a bunch of particles with given momentum together and more particles come out.
因此，散射振幅基本上是概率，它告诉您如果将一堆具有给定动量的粒子扔在一起并出现更多粒子，则可能会发生什么。
Well, I guess that sort of classical approach to scattering amplitudes was to use some complicated diagrams called Feynman diagrams.
嗯，我猜这种散射振幅的经典方法是使用一些称为费曼图的复杂图表。
But the physicist Nima Arkani-Hamed and collaborators realized that there were more compact ways to understand these scattering amplitudes.
但物理学家 Nima Arkani-Hamed 和合作者意识到，有更紧凑的方法来理解这些散射幅度。
And they involved a lot of the machinery of the positive Grassmannian.
它们涉及很多积极的格拉斯曼主义机制。
Um, yeah.
嗯，是的。
And then this in turn led to a beautiful geometric object that they call the amplituhedron, whose volume computes scattering amplitudes.
然后这又产生了一个美丽的几何物体，他们称之为振幅面体，其体积计算散射振幅。
Before we start delving into the amplituhedron, if I’m saying that right, I, there was one question I had about something in, in doing a little background reading that you mentioned, these Feynman diagrams.
在我们开始深入研究振幅面体之前，如果我说得对的话，我对你提到的这些费曼图进行一些背景阅读时有一个问题。
It’s a wonderful technique for calculating the kinds of information that physicists need to try to match what they see in their experiments or to make predictions about future experiments.
这是一种很棒的技术，可以计算物理学家需要的信息类型，以尝试匹配他们在实验中看到的内容或对未来的实验进行预测。
But it can be very arduous.
但这可能非常艰巨。
There could be thousands of diagrams, sometimes even more that they have to calculate on computers.
可能有数千个图表，有时甚至更多，他们必须在计算机上计算。
And the crazy thing that seems to have come out in the amplituhedron story, as done by the physicists, is that the thousands of calculations can be reduced sometimes to one calculation.
正如物理学家所做的那样，在幅面体故事中似乎出现的疯狂的事情是，数千次计算有时可以减少为一次计算。
That is, when you mention calculating a volume, it’s analogous to finding a volume of a shape.
也就是说，当您提到计算体积时，它类似于查找形状的体积。
And it seems like a miracle.
这似乎是一个奇迹。
How could a thousand or a million things be replaced by one thing?
一千个、一百万个东西怎么可能被一个东西代替呢？
And it reminded me of cancellations that I teach when I teach calculus.
这让我想起了我教微积分时取消的教学。
There’s something we teach and you probably have to teach calculus from time to time too.
我们教一些东西，你可能也需要时不时地教微积分。
We talk, call it a telescoping series where there’s a series of terms and then on the inside there’s a lot of things being added and then subtracted again and added and subtracted and they all collapse.
我们说，称之为伸缩级数，其中有一系列项，然后在内部有很多东西被添加，然后再次减去，添加和减去，然后它们都崩溃了。
And I feel like from what I read, that in your picture you, ’cause we talked about positive as an adjective applied to the Grassmannian, that when you have this positivity extra thing thrown in there, it somehow gives rise to this kind of, it’s not the same cancellation as in a telescoping series, but it feels like it has that flavor.
从我读到的内容来看，我觉得你的照片里就是你，因为我们谈到积极作为一个形容词应用于格拉斯曼式的，当你把这种积极的额外东西扔在那里时，它会以某种方式产生这种，它与伸缩系列中的取消不同，但感觉它有那种味道。
A lot of internal cancellation simplifying a big messy thing to something much simpler.
大量的内部取消将一个大而混乱的事情简化为更简单的事情。
Am I on the right track with that?
我的方向正确吗？
I mean, even morally, if not in detail.
我的意思是，即使在道德上，如果不是细节的话。
So, there are many cancellations that occur when one goes from kind-of Feynman diagram expressions to the sort of most compact expressions that we know.
因此，当人们从一种费曼图表达式转变为我们所知的一种最紧凑的表达式时，会发生许多取消。
A big advance in this area was the recurrence of BCFW, Britto, Cachazo, Feng, and Witten, and they wrote down this beautiful and much more compact recurrence for computing scattering amplitudes.
该领域的一大进步是 BCFW、Britto、Cachazo、Feng 和 Witten 的递归式，他们写下了这个漂亮且更紧凑的递归式，用于计算散射振幅。
And then what was noticed a few years later by a physicist named Hodges was that in some special cases, if you take the recurrence and you express your amplitude as a sum of terms, it looked like the sum of terms was computing the volume of some geometric object by cutting it into pieces and adding up the volumes of those pieces.
几年后，一位名叫霍奇斯的物理学家注意到，在某些特殊情况下，如果采用递推式，并将振幅表示为项之和，看起来项之和就是通过将某个几何对象切成碎片并将这些碎片的体积相加来计算其体积。
So, this was an observation of Hodges in a few very special cases, and then he asked the question, “Is this true in general?” Can we write all of these scattering amplitudes as computing volumes of some geometric object by cutting them up into pieces and summing them up?
所以，这是霍奇斯在一些非常特殊的情况下的观察，然后他问了一个问题：“一般情况下这是真的吗？”我们能否将所有这些散射幅度写成通过将某些几何对象分割成碎片并求和来计算其体积？
So Nima Arkani-Hamed and Jaroslav Trnka invented/discovered the amplituhedron as the answer to this question.
因此，Nima Arkani-Hamed 和 Jaroslav Trnka 发明/发现了幅面体作为这个问题的答案。
So they defined this object, and it’s closely related to the positive Grassmannian, and they proposed in their 2013 paper that this was the answer to Hodges’s question.
所以他们定义了这个物体，它与正格拉斯曼密切相关，他们在2013年的论文中提出，这就是霍奇斯问题的答案。
The volume of this object is indeed computing the scattering amplitudes in question.
该物体的体积确实正在计算所讨论的散射幅度。
So, maybe we should close our discussion here by just going into a little bit of what you’ve been doing very recently, in connection with a project known as First Proof.
所以，也许我们应该通过简单介绍一下您最近所做的与“First Proof”项目相关的事情来结束我们的讨论。
Can you fill us in on what this project is about and what you’re trying to do with it?
您能否向我们介绍一下这个项目的内容以及您想用它做什么？
Yeah.
是的。
So First Proof is a project that we initiated in the fall, and the motivation and the idea was to try to come up with an objective measure of how good AI systems are at coming up with proofs of mathematical statements.
因此，“First Proof”是我们在秋季启动的一个项目，其动机和想法是试图提出一个客观的衡量标准，衡量人工智能系统在提出数学陈述证明方面的表现。
There’s been a lot of noise in the media either sort-of hyping up the ability of AI or denigrating it, and we thought mathematicians themselves should try to figure out how best we can use AI in our own research, and in particular, to figure out how good AI is at coming up with proofs of statements.
媒体上有很多噪音，要么是炒作人工智能的能力，要么是诋毁它，我们认为数学家自己应该尝试弄清楚如何在我们自己的研究中最好地使用人工智能，特别是弄清楚人工智能在提出陈述证明方面有多好。
But this is a very tricky thing to test because LLMs, AI models are extremely good at searching the literature.
但这是一件非常棘手的事情，因为法学硕士、人工智能模型非常擅长搜索文献。
So, if you ask your favorite AI model to come up with a proof of a mathematical statement, if that statement and proof are on the internet somewhere, it’s gonna find it.
所以，如果你要求你最喜欢的人工智能模型提出一个数学陈述的证明，如果该陈述和证明在互联网上的某个地方，它就会找到它。
So we wanted to know how good is it at coming up with new proofs that aren’t already out there.
所以我们想知道它在提出尚未存在的新证据方面有多好。
And so what we decided we needed to do was take mathematical statements, lemmas, say from our own research, where we had proved the lemma or the statement, but we had not released the solution on the internet anywhere, and propose these kinds of statements as problems, as a challenge for AI systems.
因此，我们决定需要做的是从我们自己的研究中获取数学陈述、引理，我们已经证明了引理或陈述，但我们没有在互联网上的任何地方发布解决方案，并将这些陈述作为问题提出，作为对人工智能系统的挑战。
So, a group of 11 of us got together and produced these kinds of problems from our work and put them out on the internet in a paper on February 6 as a challenge for AI systems.
因此，我们 11 个人聚集在一起，从我们的工作中提出了此类问题，并于 2 月 6 日将其作为对人工智能系统的挑战以论文形式发布到互联网上。
That’s February 6th, 2026 for people in the future listening to this.
对于未来的人们来说，这是 2026 年 2 月 6 日。
That’s, that’s right.
就是这样，没错。
Yes.
是的。
And then what we did at the time was we wanted to sort-of make clear that we had solved these problems ourselves.
然后我们当时所做的就是我们想表明我们已经自己解决了这些问题。
We encrypted our solutions, we put the encrypted solutions on the internet, and then we said that we would release the key to the encryption, we’d release the solutions, publicly in one week’s time.
我们加密了我们的解决方案，我们将加密的解决方案放在互联网上，然后我们说我们将发布加密密钥，我们将在一周内公开发布解决方案。
And so during that time, we were really gratified to see that there was just an incredible amount of interest, both from the mathematical community, like professional mathematicians or math afficionados, but also from the big AI companies, you know, jumping on the challenge and seeing what they could do.
因此，在那段时间里，我们真的很高兴地看到，数学界（如专业数学家或数学爱好者）以及大型人工智能公司都表现出极大的兴趣，他们纷纷迎接挑战，看看他们能做什么。
Yeah.
是的。
’cause these are not like the Olympiad problems or the high school math contest problems or anything like that.
因为这些不像奥林匹克问题或高中数学竞赛问题或类似的问题。
These are really research-level questions, but bite-sized.
这些确实是研究级别的问题，但规模很小。
As you say, they’re lemmas, not the whole paper.
正如你所说，它们是引理，而不是整篇论文。
Right, right, right.
对，对，对。
So this was a new kind of challenge because as you said, most previous benchmarks consisted of problems with numerical answers, as opposed to answers that consisted of proofs.
所以这是一种新的挑战，因为正如您所说，大多数以前的基准测试都包含数字答案的问题，而不是包含证明的答案。
So with all of our problems, we made sure that we had proofs that were roughly five pages in length, or less.
因此，对于我们所有的问题，我们确保我们拥有大约五页或更短的证明。
And how did the AIs do?
人工智能表现如何？
Is it possible to assess?
可以评价吗？
Yeah, so we did our own private assessments at the time that we came up with these 10 questions.
是的，所以我们在提出这 10 个问题时进行了自己的私人评估。
And, actually deciding the protocols around testing is also a tricky thing to do because you could give an AI model one shot to answer the question.
而且，实际上决定测试的协议也是一件棘手的事情，因为你可以给人工智能模型一次机会来回答这个问题。
You know, you could just give it the problem and see how it does.
你知道，你可以给它提出问题，看看它是如何运作的。
Or one could have an extended conversation with the model and try to coax it to give a better answer.
或者，人们可以与模型进行长时间的对话，并尝试诱使其给出更好的答案。
But so in our private tests that we did beforehand, we just gave each AI model one shot to answer the question.
但在我们之前进行的私人测试中，我们只是给每个人工智能模型一次机会来回答这个问题。
We didn’t have any back and forth, and what we found at that time was that the models could solve two of our 10 questions.
我们没有任何来回，当时我们发现这些模型可以解决我们 10 个问题中的两个。
Oh, okay.
哦好的。
That’s not bad.
那还不错。
These are hard questions.
这些都是难题。
Yeah, yeah, yeah.
是啊，是啊，是啊。
No. Not bad.
不，不错。
And during that week various individuals and also people with the companies were working on the problems and coming up with solutions.
在那一周里，各种个人和公司人员都在研究问题并提出解决方案。
And if you sort-of put together the best efforts from all of the different people and groups who submitted answers, we did get perhaps correct solutions to six of the 10.
如果你把提交答案的所有不同的人和团体的最大努力放在一起，我们确实得到了 10 个答案中的 6 个的也许正确的解决方案。
But we are trying to shy away from making any formal statements about how people or groups did because we didn’t lay any ground rules.
但我们试图避免对人们或团体的表现做出任何正式声明，因为我们没有制定任何基本规则。
Since different people and different groups and different companies would’ve had different procedures, and different amounts of feedback, it’s hard to sort of compare how the models did.
由于不同的人、不同的群体和不同的公司会有不同的程序和不同数量的反馈，因此很难比较模型的表现。
And so now you have very recently, it was only a few days before our conversation right now, you released what you’re calling, what are you calling it?
所以现在你最近，就在我们现在谈话的前几天，你发布了你所说的东西，你称之为什么？
Yes.
是的。
First Proof is a baking pun.
First Proof 是一个烘焙双关语。
It’s about proofing the dough before you bake it.
这是关于在烘烤面团之前对面团进行校样。
And so we put out, you know, our first batch of problems back in February and just a few days ago on March 14, 2026, on Pi Day, we put out an announcement that we will release a second batch of problems sometime later in the spring.
因此，我们在 2 月份就发布了第一批问题，就在几天前，即 2026 年 3 月 14 日，圆周率日，我们发布了一项公告，称我们将在春季晚些时候发布第二批问题。
They will similarly be sort of bite-sized problems from different areas of mathematics coming from research of mathematicians.
它们同样是来自数学家研究的不同数学领域的小问题。
But this time we mean for our problems to be a more formal benchmark.
但这一次我们的意思是让我们的问题成为一个更正式的基准。
And we do intend to get the solutions graded at the end.
我们确实打算在最后对解决方案进行评分。
Okay.
好的。
Well, this’ll be interesting to see.
嗯，这会很有趣。
Are there any discoveries about either of the things we really talked about the Grassmannian and its relatives, or this AI work, you most hope to see, say 10 years from now?
关于我们真正谈论的格拉斯曼及其亲属的事情，或者你最希望看到的人工智能工作，比如 10 年后，有什么发现吗？
As far as the Grassmannian goes, I’m hopeful that maybe there’s even more exciting connections to other parts of the real world.
就格拉斯曼而言，我希望也许与现实世界的其他部分有更令人兴奋的联系。
And as far as the AI model go, it’s very hard for me to predict.
就人工智能模型而言，我很难预测。
You know, it feels like the ecosystem in which we’re doing math is being upended and we’re trying to figure out how best to adapt, how we can use these new tools.
你知道，感觉我们做数学的生态系统正在被颠覆，我们正在试图找出如何最好地适应，如何使用这些新工具。
I would hope that 10 years from now, they would be sort of research partners, with a sort of higher level of reliability and confidence than we have at the moment.
我希望 10 年后，他们将成为某种研究合作伙伴，比我们现在拥有更高水平的可靠性和信心。
All right, and the last thing, is there something you could put your finger on that particularly is a source of joy for you as a mathematician?
好吧，最后一件事，作为一名数学家，有什么事情是你可以特别高兴的吗？
What brings you joy in your work?
是什么给你的工作带来快乐？
There’s been a lot of noise in the media either sort-of hyping up the ability of AI or denigrating it, and we thought mathematicians themselves should try to figure out how best we can use AI in our own research, and in particular, to figure out how good AI is at coming up with proofs of statements.
媒体上有很多噪音，要么是炒作人工智能的能力，要么是诋毁它，我们认为数学家自己应该尝试弄清楚如何在我们自己的研究中最好地使用人工智能，特别是弄清楚人工智能在提出陈述证明方面有多好。
I think it’s identifying connections between things I didn’t expect to be connected.
我认为它正在识别我没想到的事物之间的联系。
You know, just finding these kinds of connections, whether it’s to the traffic flow, or to the shallow water waves, or to the scattering aptitudes.
你知道，只要找到这些类型的联系，无论是交通流量，还是浅水波浪，还是散射能力。
I have so many stories from my research where I might have a conversation with another mathematician and they show me some numbers of something that they were computing, and then I recognize them as having come up before.
我的研究中有很多故事，我可能会与另一位数学家进行对话，他们向我展示了他们正在计算的一些数字，然后我认出它们以前曾出现过。
It’s always so exciting and intriguing.
它总是那么令人兴奋和有趣。
I mean, it’s this sort of mystery and then we have to do the detective work of figuring out how these objects are connected.
我的意思是，这是一个谜，然后我们必须进行侦探工作，弄清楚这些对象是如何连接的。
Yeah, so I think that’s the thing that I find most exciting.
是的，所以我认为这是我觉得最令人兴奋的事情。
And then of course, the joy is when you realize, you make that connection.
当然，当你意识到你建立了这种联系时，你就会感到高兴。
You understand, you have this realization of how they are secretly connected and how you can sort of make that rigorous.
你明白，你已经意识到它们是如何秘密连接的，以及如何使其变得严格。
Well, very, very good.
嗯，非常非常好。
It’s really been fun.
这真的很有趣。
Thank you, Lauren.
谢谢你，劳伦。
Wow.
哇。
So this is terrifying, right?
所以这很可怕，对吧？
Now, I really do wonder, hey, have I written the last of my very, very technical papers.
现在，我真的想知道，嘿，我是否写了最后一篇非常非常技术性的论文。
But then I also remember the time that computers were first invented, and everybody was saying …
但后来我也记得计算机首次发明的时候，每个人都在说……
that’s not…
那不是……
I don’t remember when computers were first invented.
我不记得计算机是什么时候第一次发明的。
But you know what I’m saying.
但你知道我在说什么。
When they got cheaper, more readily available, large processing machines could do huge datasets, and the same kind of thing was said: “Well, now the technical people are obsolete.” I don’t know.
当它们变得更便宜、更容易获得时，大型处理机器可以处理巨大的数据集，同样的事情也有人说：“好吧，现在技术人员已经过时了。”我不知道。
What do you think?
你怎么认为？
Well, um, I’m confused about it.
嗯，嗯，我对此感到困惑。
I really am of two minds, and, you know, we still have CAPTCHA, that thing where you have to identify that you’re not a robot by doing some little image processing.
我确实有两种想法，而且，你知道，我们仍然有验证码，你必须通过进行一些小的图像处理来识别你不是机器人。
Apparently, that’s still hard for the AIs.
显然，这对于人工智能来说仍然很难。
So yes, they’re very good at certain things, but there’s still a way to go.
所以，是的，他们在某些事情上非常擅长，但还有很长的路要走。
They also seem to lack common sense in a lot of domains.
他们似乎在很多领域也缺乏常识。
But still, back to your question though, I mean, will they make you and me and people like us obsolete?
但是，回到你的问题，我的意思是，它们会让你和我以及像我们这样的人变得过时吗？
Because what we do isn’t exactly the realm of common sense.
因为我们所做的并不完全符合常识。
We’re in a…
我们在……
as my wife would be the first to tell you.
我的妻子会第一个告诉你。
Right, exactly.
对，完全正确。
And she’d be right.
她是对的。
No, but you know, like a lot of the work we do involves very explicit rules.
不，但是你知道，就像我们所做的很多工作都涉及非常明确的规则一样。
You could imagine we might be at risk more than the people who do plumbing or caregiving, who will be the last to be superseded by robots and AIs.
你可以想象，我们可能比那些从事管道或护理工作的人面临更大的风险，他们将是最后被机器人和人工智能取代的人。
Well, just to play devil’s advocate, I think the idea of these machines as thought partners is closer to what I’m imagining is going to happen, because I still don’t see the machine asking the questions.
好吧，只是唱反调，我认为这些机器作为思想伙伴的想法更接近我想象中将会发生的事情，因为我仍然没有看到机器提出问题。
Not yet, no. Do you think in the era when AI starts doing math alongside us, or maybe even instead of us, will beauty play the same role then?
还没有，没有。你认为当人工智能开始与我们一起做数学的时代，或者甚至代替我们做数学时，美会扮演同样的角色吗？
Like a guide to what you should think about, what questions you should ask, how to judge whether you’re on the right track with the theorems you can obtain.
就像指导你应该思考什么、你应该问什么问题、如何用你可以获得的定理来判断你是否走在正确的轨道上一样。
Gosh, it’s a really…
天哪，这真是一个……
profound question, ’cause one of the roles beauty might be playing is rendering some very complex subject comprehensible to us.
这是一个深刻的问题，因为美可能扮演的角色之一就是让一些非常复杂的主题为我们所理解。
Which I really need because I don’t have infinite compute.
我确实需要它，因为我没有无限的计算能力。
So, I need to have a more aesthetic approach.
所以，我需要有一种更美观的方法。
So, I mean, you could kind of say, in a way, nature already has all the answers.
所以，我的意思是，你可以说，在某种程度上，大自然已经有了所有的答案。
The whole game is discovering what nature already knows.
整个游戏正在探索大自然已经知道的事情。
So, if the AI just simply has this infinite list of things it knows, you know, if we don’t understand it, I don’t know that the game has changed that much.
所以，如果人工智能只是简单地拥有它所知道的无限的事物列表，你知道，如果我们不理解它，我不知道游戏已经改变了那么多。
I don’t know.
我不知道。
What do you think?
你怎么认为？
I always wonder about is understanding overrated?
我总是想知道理解是否被高估了？
So, here’s what I mean, that we might be confusing means and ends.
所以，这就是我的意思，我们可能会混淆手段和目的。
Like, if the end is to predict nature, to be able to find formulas and theorems that are true, understanding may be a crutch.
就像，如果最终的目的是预测自然，能够找到正确的公式和定理，那么理解可能是一个拐杖。
It helps us get good answers.
它可以帮助我们得到好的答案。
It helps us get more control over the universe, but it’s not the game.
它帮助我们更好地控制宇宙，但这不是游戏。
Like, if you’re trying to save someone’s life, you may have to come up with a medical therapy that you don’t understand that works.
就像，如果你想拯救某人的生命，你可能必须想出一种你不知道有效的药物疗法。
And so it’s not always so clear to me that understanding is the goal in itself.
因此，我并不总是那么清楚理解本身就是目标。
But on the other hand, there are people who say it’s not science without understanding.
但另一方面，也有人说不理解就不是科学。
It’s something less than science.
这不是科学。
It’s like a degradation of the human spirit.
这就像人类精神的退化。
Why even do it if you’re not understanding?
如果你不理解，为什么还要这么做呢？
I don’t know what to think about that.
我不知道对此有何看法。
I can see both sides of that argument.
我可以看到这个论点的双方。
But what’s really interesting in what Lauren Williams and her colleagues are doing is they are giving these secret problems from research-level math that haven’t been published, so the AI can’t look them up on the internet, and asking them how many of our 10 problems can you solve?
但劳伦·威廉姆斯和她的同事所做的事情真正有趣的是，他们给出了这些尚未发表的研究级数学秘密问题，因此人工智能无法在互联网上查找它们，并询问他们你能解决我们的 10 个问题中的多少个问题？
It’s just an interesting benchmark, different methodology than we’re seeing elsewhere.
这只是一个有趣的基准，与我们在其他地方看到的方法不同。
Yeah.
是的。
Yeah, it’s amazing ’cause it means they’re not just regurgitating, culling a human response.
是的，这太神奇了，因为这意味着它们不仅仅是反刍、剔除人类的反应。
So far they’re not mastering that.
到目前为止他们还没有掌握这一点。
They’re not climbing the whole mountain.
他们并没有攀登整座山。
But right now, in the context of what people are doing, is it possible to have a machine that says, “You know, here’s an interesting idea,” or, you know, “I’m bored today I’m going to try this,” or…
但现在，在人们正在做的事情的背景下，是否有可能有一台机器说，“你知道，这是一个有趣的想法，”或者，你知道，“我今天很无聊，我要尝试这个，”或者......
We’ll really know that they’ve arrived when they’re a guest on The Joy of Why .
当他们作为《The Joy of Why》的嘉宾时，我们才会真正知道他们已经到来。
Yeah.
是的。
When we have Claude on.
当克劳德上场的时候。
Yeah, when we have Claude, and course, by then, maybe we won’t be the hosts anymore.
是的，当我们有了克劳德，当然，到那时，也许我们就不再是东道主了。
See you later, Janna.
再见，詹娜。
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PRX Productions 的执行制片人是乔斯林·冈萨雷斯 (Jocelyn Gonzales)。
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特别感谢康奈尔广播工作室的 Garth Avery。
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我是你们的主持人，史蒂夫·斯特罗加茨。
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Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.
《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.
辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.
版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。