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第 22 期 · 回应 Ⅱ·02「怎样才算真正学会？」
Terry Tao "How to think like a mathematician" presented by the UCLA Curtis Center
UCLA Curtis Center章节 · 点击跳转视频
0:00
圆周率日开场与陶哲轩介绍
1:59
公众对数学家的三种误解
4:31
前严格、严格、后严格三阶段
6:31
反证法：从课间游戏到哈代的弃着
11:06
数学是失败最廉价的学科
14:42
只发表胜利的文化与十三项的教训
17:20
问蠢问题：斯坦不等式的反例
20:50
协作、分工与数学的工业革命
24:24
大语言模型的真实成功率
27:59
神童教育与犯错的特权
33:25
AI 能否创造定义与阶段是否必然
38:57
卡在同一扇门前：做数学的感觉
本期讲者
陶哲轩UCLA 数学系教授、James and Carol Collins 讲席教授，2006 年菲尔兹奖得主。10 岁参加国际数学奥林匹克，24 岁成为 UCLA 史上最年轻正教授，已出版 16 本书，研究横跨调和分析、偏微分方程、组合数学与解析数论。
希瑟·达拉斯UCLA 柯蒂斯中心（Curtis Center for Mathematics and Teaching）主任，主持演讲后的讨论环节，研究方向为数学教育与教师发展。
凯特·史蒂文森加州州立大学北岭分校（CSUN）数学系主任，亲自讲授本科证明入门课，关注第一代大学生的数学学习与信任建立。
01圆周率日开场与陶哲轩介绍
0:00
One thing that we'll want to think about is this question of how do we get students to really think hard about these mathematical concepts. What we're going to do is we're going to have a speaker who will give us some food for thought. I guess it's Pi Day, so pie for thought if you will. And we'll have Heather Dallas, the director of the Curtis Center, to come up afterwards and lead us in this conversation. So I'm hoping that you'll listen intently to the words that are going to be said, and that we'll have a chance to discuss and have an interchange of ideas afterwards.
我们要思考的一个问题是，怎样才能让学生真正深入地思考这些数学概念。我们接下来要请一位演讲者，给我们带来一些值得思考的东西。我想今天是圆周率日（Pi Day），所以不妨说是「派」（pie）给大家带来的思考。之后我们会请柯蒂斯中心（Curtis Center）的主任 Heather Dallas 上台，带领我们展开这场讨论。所以我希望大家能请专心听接下来要讲的内容，之后我们会有机会讨论、互相交流想法。
0:34
That being said, I do want to introduce our speaker for today, who's a professor in the Department of Mathematics and the James and Carol Collins Chair in the College of Letters and Sciences. Um I want to give you 3.14 interesting facts about Professor Tao. Um interesting fact number one, he is the youngest participant to date in the International Math Olympiad, first competing at the age of 10. Fact number two, when he was 24, he was promoted to full professor at UCLA and still remains the youngest person ever appointed to that rank by UCLA.
说完这些，我确实想介绍一下今天的演讲者，他是数学系的教授，也是文理学院的詹姆斯与卡罗尔·柯林斯讲席教授。（文理学院）嗯，我想跟大家分享关于陶教授的 3.14 个有趣的事实。嗯，有趣的事实之一：他是国际数学奥林匹克竞赛至今为止最年轻的参赛者，10 岁就首次参赛了。事实之二：他 24 岁时就被提升为加州大学洛杉矶分校的正教授，至今仍是 UCLA 有史以来获此职称最年轻的人。
1:11
Fact number three, Professor Tao has published 16 books, but he has a new book coming out, Six Math Essentials, that'll be his first popular math book. And interesting fact 0.14 is that today he'll give us a talk entitled "What Does It Mean to Think Like a Mathematician?" So let's give him a round of applause. >> [applause] >> Thank you, Andre. That's that's a great introduction and I'm very happy to be here. Uh technically I think I was a colleague of Phil Curtis. I came here in '96. Uh it was about the same time he was retiring. I'm not quite sure exact dates. Um so I never actually got to interact with him too much, I'm afraid. Um but he was a living legend. Well, he was a legend for our department.
事实之三：陶教授已经出版了 16 本书，而他还有一本新书即将面世，书名是《Six Math Essentials》，这将是他的第一本数学科普读物。有意思的一点是 0.14，今天他要给我们做一场演讲，题目是《像数学家一样思考意味着什么？》让我们用掌声欢迎他。>> [掌声]>> 谢谢你，Andre。这个介绍非常棒，我很高兴能来到这里。呃，严格来说，我算是Phil Curtis 的同事。我是 1996 年来到这里的。呃，那大概正是他退休的时候。我不太确定具体日期。嗯，所以恐怕我从来没有和他有太多的交流。嗯，但他是一位在世的传奇人物。嗯，他是我们系的传奇。
02公众对数学家的三种误解
1:59
Um so yeah, so I'm here to sort of bring the perspective of a mathematician to to this event. And I think it you know, it's it is important we have the way of thinking like a mathematician, I really enjoy it. Um and it's it's a very precious thing that wish more uh students would would gain access to. But it's just it's it's often such a shame. Um and it's I think, you know, most people don't have a handle of even what a mathematician does. You know, so you know, maybe a lawyer or doctor or engineer, you have some idea of what they do. Uh a mathematician, it is it is a little bit uh the the mental images people have in their heads are a bit a bit uh inaccurate.
嗯，所以是的，我来这里是想为这个活动带来一位数学家的视角。我觉得，你知道的，拥有像数学家一样的思维方式是很重要的，我非常享受这种方式。嗯，而且这是一件非常宝贵的事情，我希望有更多的学生能够接触到它。但常常，这真的挺遗憾的。嗯，而且我觉得，你知道，大多数人甚至都不了解数学家到底是做什么的。你知道的，比如说，律师、医生或者工程师，你多少能想象出他们在做什么。呃，但数学家嘛，这就有点儿，呃，人们脑子里的那些印象其实是有点儿，有点儿，呃，不太准确的。
2:39
And uh you know, every mathematician has had this experience that it's very hard it's very awkward at parties. You know, like you you I ask, "What what do you do?" And so I'm a mathematician, and you always get this response, "Okay, I'm bad at math at school." Um well, actually sometimes you get like a really enthusiastic response, but this that's that's that's the median response. Um Some people think that mathematicians are like wizards, that that we gain access to these magic spells and you know, and with these weird symbols that that we can somehow cast on people. Um Some people think that it's really complicated technical skill that and there's only one correct answer and lots and lots of wrong answers and you make one mistake the whole thing collapses and we are somehow juggling all kinds of complicated equations.
而且，你知道，每个数学家都有过这样的经历，那就是在派对上这非常难、非常尴尬。你知道的，就像有人问：“你是做什么的？”然后我说我是数学家，你总会得到这样的回应：“好吧，“我数学不好。”嗯，其实有时候你也会得到特别热情的回应，但那个……那个是中位数的反应。嗯，有些人觉得数学家像巫师一样，觉得我们掌握了某些魔法咒语，你知道的，用那些奇怪的符号，好像能对人施法一样。嗯，有些人觉得这是一种非常复杂的技术性技能，只有一个正确答案，还有非常非常多的错误答案，只要犯一个错，整件事就崩塌了，而我们不知怎的能同时玩转各种复杂的方程。
3:26
Um Or you know, they think that we we we are you know, like these Hollywood, you know, sort of we see all these equations and I we don't or maybe some people do, but I don't certainly. We don't see these things in in front of our in in in front of our eyes when we uh when we do our math. Um so we we have all these sort of uh um um archetypes which I'm not the norm really. I mean, most of us are pretty normal, I think. Um oops Um so which is unfortunate in many ways. Um I think, you know, I mean, on the one hand it is a little bit cool that you know, sometimes we we get this sort of reputation for being geniuses and and so forth, but it it does create um um you know, issues with students, you know, that that you know, that maybe they they they read stories about mathematicians solving these great problems and being very clever, and then they have their own math problem to solve and they and they can't solve it in one go and um you know, all all they um yeah, yeah, or they they they they learn a how to
嗯，或者你知道的，他们觉得我们就像好莱坞演的那样，眼前会浮现出所有这些方程而我们并没有，或者也许有些人有，但我肯定没有。我们做数学的时候，眼前并不会浮现这些东西。嗯，所以我们身上有各种各样的刻板印象，而我其实并不是那种典型。我是说，我们大多数人都挺正常的，我觉得。嗯哎呀。嗯，所以这在很多方面都挺遗憾的。嗯，我觉得，你知道，一方面这也有点酷，就是我们有时候会被当成天才之类的，等等，但这确实会造成一些问题，比如对学生来说，你知道的，他们可能读到一些关于数学家解决那些伟大难题、非常聪明的故事，然后轮到他们自己要解一道数学题，一次做不出来，然后他们，嗯，他们就……是啊，是啊，或者他们在学一个很难的题目，怎么都学不会。于是他们就
03前严格、严格、后严格三阶段
4:31
learn a difficult topic and they never get it. And so they they they quit math before before it becomes fun. Um Yeah, and so you know, if if you teach that mathematicians are happy geniuses, then and and you don't feel like a genius, then you don't feel like a mathematician. Um so I just wanted to sort of share what a little bit of what it's like to feel like be a mathematician. This is something every mathematician knows about, but we don't communicate it as often as we should. I think we we should do more outreach operation. Um and we we do sort of hang out in our ivory tower maybe a bit too much. Um So um one thing I I um one of my sort of frameworks for thinking like a mathematician is that is that there's actually multiple stages of thinking like a mathematician. Uh and that's that's part of why math is um mathematical thought is a bit hard to uh to internalize because it is it is a somewhat complex um thing to evolve. Um So I like to divide math into what I call the pre-rigorous, rigorous, and rigorous stages. Um so
在数学变得有趣之前放弃了。嗯，是啊，所以你知道，如果你教给别人的是数学家都是快乐的天才，那么当你觉得自己不是天才时，你就会觉得自己不是当数学家的料。嗯，所以我只是想稍微分享一下，做一个数学家是什么感觉。这是每个数学家都知道的事，但我们没有像应该的那样经常去讲。我觉得我们应该多做一些科普推广工作。嗯，而且我们确实有点太待在象牙塔里了。嗯，所以，有一件事我，嗯我思考“像数学家一样思考”的框架之一是，其实“像数学家一样思考”是分好几个阶段的，嗯，这也是为什么数学思维有点难以内化的部分原因，因为它是一个相当复杂的、需要逐步演化的东西。嗯，所以我喜欢把数学分成我称之为“前严格”、“严格”和“后严格”的阶段。嗯，前严格阶段大致是K12、K14那种教育阶段，你被教的是
5:32
pre-rigorous is roughly sort of K12, K14 type um of of education where you're taught examples, intuition, formulas, computing, um but it's but you can make mistakes and you don't really understand what's going on. Um you just have sort of a vague handle on everything. Um And then you go to to college, um and if you're if you're in one of the math majors, you'll get taught these dreaded proof classes where you start learning how to think rigorously. And and now there's only one correct way to do to solve problems and lots and lots of incorrect ways and a lot of your old pre-rigorous intuition gets sort of pooh-poohed and they say, "No, no, that that that was for kids. Now now this is this is the real math." Um But then what is not really appreciated because most people don't reach the stage is that actually there's a third stage, which I call the post-rigorous stage, where once you know how to do these things rigorously and and precisely, you actually go back to revisit your intuition and and think
例子、直觉、公式、计算，嗯，但你会犯错，而且你并不真正理解到底是怎么回事。嗯，你对一切只有一个模糊的把握。嗯，然后你上了大学，嗯，如果你读的是数学专业，你就会上那些让人闻风丧胆的证明课，你开始学习如何严格地思考。而现在解题只有一种正确的方式，还有非常非常多的错误方式，而且你以前那些前严格的直觉会被嗤之以鼻，他们会说：“不不不，那是给小孩子的。现在这才是真正的数学。”嗯，但有一点没有得到充分重视，因为大多数人到不了那个阶段，那就是其实还有第三个阶段，我称之为“后严格”阶段，在这个阶段，一旦你知道如何严格、精确地做这些事，你其实会回过头去重新审视你的直觉，然后
04反证法：从课间游戏到哈代的弃着
6:31
much more fluidly and informally, but knowing that now if you wave your hands and do something, you can convert it to a rigorous argument if if if you want and you can go back and forth. Um and this is basically graduate school. Um and that's the fun part and it's a shame that most people don't see that. Um so that's the word. Let me try to illustrate with sort of examples of what these phases look like. So let me talk for example, there's a basic concept in math called proof by contradiction. Um and it's considered sort of a not unintuitive and difficult concept for for students to to understand, but actually um um um primary school kids teach teach this concept to themselves in fact at you know, at recess. For example, you know, I mean, when I was a kid we had silly games like this. We would we would gather around and we would just name you know, who can name the biggest number? You know, and so I would name 1 billion, 1 trillion, 1 quadrillion, and you just go back and forth and 1 1 trillion trillion trillion, whatever. Um
以更加流畅、更加非正式的方式思考，但你知道，现在如果你挥挥手做点什么，只要你想，你就能把它转化成一个严格的论证，你可以在两者之间来回切换。嗯，这基本上就是研究生阶段。嗯，那才是有趣的部分，可惜大多数人看不到这一点。嗯，这就是我想说的。让我试着举些例子来说明这些阶段是什么样子。比如说，数学里有一个基本概念叫反证法。嗯，它通常被认为是一个不太直观、对学生来说比较难理解的概念，但其实，嗯，小学生在课间休息的时候就自己教会了自己这个概念。比如说，你知道，我小时候我们玩过这样的傻游戏。我们会围在一起，就比谁能说出最大的数字，你知道吧？我会说十亿，一万亿，一千万亿，然后就这样来回，一万亿万亿万亿，随便什么。嗯，这个游戏会一直玩下去，直到有人意识到，哎呀，不管
7:34
And this game would go on until someone realizes um oops that no matter what number um someone names, the next person can always name that number plus one. Um so someone eventually figures this out. Um and at that point, they will realize there is no biggest number in in because no matter what number you can pick, there's always you can always add one and get a big number. Um and they have used proof by contradiction. This is they they proved that a biggest number cannot exist because if it did, it would be bigger than itself or be bigger than than the number plus one, which is not possible. Um And this is something that they've discovered, but it they don't they can't they can't verbalize this.
谁说出什么数字，下一个人总能说出那个数字加一。嗯，所以最终有人琢磨明白了。嗯，到那个时候，他们就会意识到，不存在最大的数字，因为不管你挑哪个数字，你总能加一得到一个更大的数。嗯，而他们已经用上了反证法。这就是，他们证明了最大的数字不可能存在，因为如果它存在，它就会比它自己大，或者说比那个数加一还大，而这是不可能的。嗯，这是他们自己发现的，但他们没法，他们说不出来。
8:14
This is pre-rigorous. They don't they don't have the language. Um So then you know, maybe you go to you take some more advanced classes in math high school and then and then undergraduate, and then you start seeing proofs. And proofs come in various flavors. Um There will be direct forward proofs where you have some word problem and you have some hypotheses and then you just start applying various mathematical transformations and you get from A to B. You get from your hypothesis to your conclusion. Um and I'm sure you've all done problems like this. I'm not going to go through that with you. Um so that's a direct forward proof.
这就是前严格阶段。他们没有那套语言。嗯，那么，你知道，也许你上了高中更高阶的数学课，然后上了本科，然后你开始接触证明。证明有各种各样的形式。嗯，有正向直接证明，就是你有一道文字题，你有一些假设条件，然后你开始运用各种数学变换，从A推到B。从你的假设推到你的结论。嗯，我相信你们都做过这类题。我就不跟你们细讲了。嗯，这就是正向直接证明。
8:48
Um There's also direct backwards proofs where you have some hypothesis conclusion. Um and now you start with what you want to prove and you reduce it, you transform it, you cancel terms, and you do all this algebra and you get back to to what you started. So this is this is very typical kind of rigorous stage mathematics where you and you're not supposed to make any mistakes as as as you do this. Um But then sometimes occasionally you're taught a proof by contradiction and it's so weird. You know, so you might be so proof that say square root of two is an irrational number, that this this number cannot be written as the ratio of two integers.
嗯，还有逆向直接证明，就是你有假设和结论。嗯，现在你从你想证明的东西出发，把它化简、变形、约掉一些项，做各种代数运算，最后回到你出发的地方。所以这是非常典型的严格阶段的数学，你在做这个过程中是不该犯任何错误的。嗯，但偶尔有时候，你会学到反证法，而它太奇怪了。你知道，比如说要证明根号二是无理数，也就是这个数不能写成两个整数之比。
9:23
And then you say, "Well, let's suppose that it is a rational number." And then you do something and you argue for a bit and then you see something say that what I the conclusion I got didn't contradict something I said earlier. Therefore, my original conclusion was true that root two is irrational. And this is really hard to for or I mean, some students get it, but definitely many students do not connect that with the kind of pre-rigorous experience they had with with contradiction that they may have um you know, learned about themselves, but it doesn't it doesn't feel like this. But it It the same concept. Um but it's taught in a very different way.
然后你说：“好，假设它是有理数。”然后你做点什么，论证一番，接着你发现某个东西，说我得到的结论和我前面说过的某句话矛盾了。因此，我最初的结论是对的，根号二是无理数。这真的很难，我是说，有些学生能懂，但肯定有很多学生没法把这个跟他们前严格阶段接触过的那种矛盾经验联系起来，那种他们可能自己领悟过的东西，但感觉完全不一样。但其实是同一个概念。嗯，只是教的方式非常不同。
10:01
So if you are so once you actually learn you become professional mathematician you use proof of contradiction proof of contradiction all the time. And it's it's not we don't view it as actually anything clever. Basically it's it's it's just one move you can make. You know, so if you doubt that something is true we just it's very natural for us to assume it's true. See what happens if there's something bad happens if if if we can get a contradiction out of it we know that that it could it couldn't be true so it has to be false.
所以，一旦你真的学会了，成为职业数学家，你就会一直用反证法，一直用。而且我们并不觉得这有什么高明的。基本上，这就是你可以走的一步棋而已。你知道，如果你怀疑某件事是不是真的，我们很自然地就会假设它是真的，看看会发生什么，如果出了什么问题，如果我们能从中推出矛盾，我们就知道它不可能是真的，所以它必然是假的。
10:33
And it's it's just a move in a game for us. So actually G. H. Hardy the mathematician made this very nice quote. So reductio ad absurdum which is Latin for proof of contradiction is one of the mathematician's finest weapons. It is a far finer gambit than a chess gambit. So a chess gambit a chess player may sacrifice a bishop or a rook to get some positional advantage but the mathematician can offer the entire game. Okay, and still win. So that's how yeah so there's three different ways of thinking about the same concept.
对我们来说这就是棋局里的一步。其实数学家G.H.哈代有一句很妙的话。reductioad absurdum，也就是拉丁文的反证法，是数学家最精良的武器之一。它是比国际象棋弃子更精妙的弃着。国际象棋棋手可能牺牲一个象或一个车来换取某种局面上的优势，但数学家可以把整盘棋都献出去。好，而且仍然能赢。所以这就是……是啊，所以对同一个概念有三种不同的思考方式。
05数学是失败最廉价的学科
11:06
Okay, so that's one of the ways one of the lessons about how to think like a mathematician. Another one which I think people already mentioned some of the previous panelists mentioned is that math is a a place where it is okay to fail which is actually the opposite of the way we teach it often especially in the rigorous phase where you know, if you if you get a sign error wrong and so forth all these all these red marks come out and you know, and and and you lose these points whatever. But actually compared to other disciplines math is actually it's very failure is very very cheap. You know, if if you're an engineer and you're designing a bridge and you make a you make a mistake that's an expensive mistake. If you're a heart surgeon and you cut the wrong thing that's a that's also a very bad mistake. But you know, if you had if you have a solving a math problem and and your proof doesn't quite work out that's a very cheap mistake. You just do it again.
好，这就是关于如何像数学家一样思考的其中一条经验。另一条，我觉得前面几位嘉宾已经有人提到过了，就是数学是一个允许失败的地方，而这其实和我们常常教它的方式恰恰相反，尤其是在严格阶段，你知道，如果你符号写错了，等等，所有这些红叉就出来了，你知道，然后你就扣分之类的。但其实，和其他学科相比，数学的失败其实非常非常廉价。你知道，如果你是工程师，你在设计一座桥，你犯了个错，那是代价高昂的错误。如果你是心外科医生，你切错了地方，那也是非常糟糕的错误。但你知道，如果你在解一道数学题，你的证明没能走通，那是非常廉价的错误。你重来一遍就行了。
11:53
Um Yeah, so the Vladimir Arnold once said that math is even you can think about it as the part of physics where experiments are cheap. Um Uh though with current AI well never mind. Okay. Um So um Yeah, so because of this as a part of what you're taught as a graduate student is just keep trying things and keep making mistakes and keep doing things even if you suspect they are likely to fail. Um because the the way in which they fail is often very valuable. And so this is a mindset that we internalize and it's just we're just used to it but like almost nobody else gets it. You know, that that that in almost any other discipline people are just afraid of making mistakes. Um but in math we have the freedom to fail and that's that's actually very precious.
嗯，是啊，所以弗拉基米尔·阿诺尔德曾说过，数学甚至可以被看作是物理学中实验很廉价的那一部分。嗯，不过考虑到现在的AI……算了，不说了。好。嗯，所以，正因为如此，你作为研究生被教导的一部分内容就是：不断尝试各种东西，不断犯错，不断去做事情，哪怕你觉得它们很可能会失败。嗯，因为它们失败的方式往往非常有价值。所以这是一种我们内化了的心态，我们就是习惯了，但几乎没有别人能理解这一点。你知道，在几乎任何其他学科里，人们都害怕犯错。嗯，但在数学里我们有失败的自由，这其实非常宝贵。
12:41
Um So it's but it it's Yeah, so um There's a disconnect between the way we we sort of assess math solutions and the way we assess math process. So solutions should be correct but the process can have lots and lots of failure and that's important. Um so I'll just give you a practical actual example. So the student came up to me a few months ago and was asking for help on a math problem. I won't tell you what the problem was but um So he had a hint. So there's a technical Taylor approximation. I don't need I don't want to say what Taylor approximation is but it's a technique that would solve the problem.
嗯，所以这……是啊，嗯，我们评价数学解答的方式和我们评价数学过程的方式之间是有脱节的。解答应该是正确的，但过程可以有非常非常多的失败，这很重要。嗯，我给你们讲一个实际的例子。几个月前有个学生来找我，问一道数学题该怎么办。我不会告诉你们是什么题，但，嗯，他有个提示。有个技术叫泰勒逼近。我不需要，我不想解释泰勒逼近是什么，但它是一种能解决这个问题的技巧。
13:19
And he had learned about Taylor approximation but he had learned it three times. So he had three different books and they said there's three different Taylor approximation formulas to use and he didn't know which one to use. And so he was paralyzed. He says I'm stuck. I don't know what to do. And I So he he experienced what someone has called analysis paralysis. There's too many options didn't know which one to pick. And I just told I said something very simple. It doesn't matter if the first thing you pick is it doesn't work. Just try one. Maybe it works maybe it doesn't work. It may partially work. Okay, but then you can see what to do next. Okay, and this is this is unblocked him. I mean I didn't give any further hints but but you know, but basically just permission to fail was was all he needed. And then I think he's off the problem. He didn't he didn't get back to me but um Um Yeah, but it's a very basic lesson which um we don't we don't often don't teach because we also because correctness in
他学过泰勒逼近，但他学过三次。他手上有三本不同的书，书上给出了三个不同的泰勒逼近公式，他不知道该用哪一个。所以他僵住了。他说我卡住了，我不知道该怎么办。而我……他经历的就是有人说的“分析瘫痪”。选项太多，不知道该选哪一个。我就跟他说了一句非常简单的话。我说，你先挑的那个行不行都没关系。就试一个。也许行，也许不行。也许只有部分管用。好，但那样你就能看出下一步该怎么走。好，这样就让他解除了阻塞。我是说，我没给任何进一步的提示，但你知道，基本上只是给他允许失败的许可，这就是他所需要的全部。然后我想他就把题做出来了。他没有再来找我，不过，嗯，是啊，这是一个非常基本的道理，嗯，我们往往不教，因为答案的正确性也很重要。所以你
14:13
the answer is important. So you have to you have to teach both. Um Yeah, but this is something that that that scientists realize. Okay, so Niels Bohr has has has this great quote that what is an expert? An expert is a person who made all the mistakes that can be made in a very narrow field. If you haven't made the certain mistakes if you haven't made them in the past you'll make them in the future. So it's actually important to make them now. So put them in the past so that that in the future you do not make that same embarrassing thing again.
必须两者都教。嗯，是啊，但这是科学家们意识到的事情。好，尼尔斯·玻尔有一句很棒的话，什么是专家？专家就是在一个非常狭窄的领域里犯过所有能犯的错误的人。如果你没有犯过某些错误，如果你过去没犯过，你将来就会犯。所以现在就把它们犯掉其实很重要，把它们放进过去，这样将来你就不会再犯同样丢人的错误了。
06只发表胜利的文化与十三项的教训
14:42
Um yeah, but it it is a really really important part of our process. Um You know, and it's it's a shame we don't report report on that. So our culture is not perfect in some way. So when we publish our papers we don't publish our papers often. I mean sometimes a few of us do when we're about all the the wrong turns and and our feeling of getting lost which is the default state of of being a mathematician actually. Um but um Yeah, but we only publish our wins usually. And then you know, you you you you read the famous mathematicians or like for example Fields Medalist Maryam Mirzakhani and you know, so I mean she's more honest than than most of us but she was but but you know, even still her papers are full of wins. And then you you read you read your own paper and you do your own work and you know, your your proofs aren't working and so forth and and you feel like an impostor.
嗯，是的，但这确实是我们工作过程中非常非常重要的一部分。嗯，你知道，很遗憾我们不会把这部分写出来。所以我们的文化在某种程度上并不完美。所以当我们发表论文时——我们并不经常发表论文。我是说，有时候我们中的少数人会写一写那些走过的弯路，以及那种迷失方向的感觉，而这其实才是做数学家的默认状态。嗯，但是，是啊，我们通常只发表我们成功的部分。然后你知道，你会去读那些著名数学家的作品，比如菲尔兹奖得主玛丽亚姆·米尔札哈尼，她其实比我们大多数人都要坦诚，但即便如此，她的论文里也全都是成功的结果。然后你再读自己的论文、做自己的研究，发现自己的证明行不通等等，于是你就觉得自己是个冒名顶替者。
15:34
And so it would be better to normalize our disclose our failures a bit more often I think. Uh oops. Um I'll just give you one example. I worked several years ago on a problem in partial differential equations. Now it's not important what the problem was. There was a very famous mathematician Jean Bourgain who worked very hard on it and we got a partial result. We wanted the whole thing. Um and so we were kind of cocky and we thought oh we'll do this in a few months. I worked with four other people and it worked. We tried something crazy and it worked. It was great. Okay, we we were in fact we were trying to book a restaurant to to celebrate with champagne. And we started writing up the proof and then one of my co-authors who was more careful than the rest of us actually noticed that you know, we expanded this thing to 13 terms and we had controlled 12 of them correctly but we just forgotten about the 13th term.
所以我认为，我们更经常地把失败正常化、公开我们的失败，会更好一些。呃，糟糕。嗯，我就举一个例子吧。几年前我研究过一个偏微分方程的问题。具体是什么问题并不重要。有一位非常著名的数学家让·布尔甘曾在这个问题上下过很大功夫，我们得到了一个部分结果。我们想要完整的结果。嗯，所以我们当时有点自负，心想，几个月就能搞定。我和另外四个人合作，而且成功了。我们尝试了一个疯狂的想法，居然奏效了。太棒了。好吧，我们实际上我们当时正准备订餐厅，打算开香槟庆祝。然后我们开始撰写证明，接着我的一位合作者——他比我们其他人都更细心——注意到，我们把这个东西展开成了13项，我们正确地控制住了其中12项，但是我们把第13项给忘了。
16:23
So oh yeah, I'll deal with this. Okay, I would add this to the paper and I checked that actually we could not control this 13th term. Actually it was actually the one term that was the worst and we had somehow dropped it. Um we thought it was a minor thing but actually it it no matter what we did we just we could not get rid of this term. And so we um Yeah, we tried all kinds of things and it and we could not fix the proof. But by that point by the time we realized that we'd spent like six months on this problem and we had and we had to cancel this reservation.
于是我说，哦好的，我来处理这个。好吧，我打算把这部分加进论文里，结果我一检查，发现我们其实根本控制不住这第13项。实际上它恰恰是最糟糕的那一项，而我们不知怎么就把它漏掉了。嗯，我们本以为这是个小问题，但实际上，不管我们怎么做，我们就是没办法消掉这一项。所以我们，嗯，是啊，我们尝试了各种办法，就是没能修补好这个证明。但等我们意识到这一点的时候，我们已经在这个问题上花了差不多六个月，我们不得不取消那个餐厅预订。
16:50
Okay, it it was we were really invested. And so we just kept at it. Whereas it would it it took us two years to solve this problem. We finally found a much different way to solve the problem. And actually this this paper is one of the papers I'm most proud of. It won a prize. You know, if we had not had this mistake of this early success we would have quit way before we had solved this problem. So actually sometimes actually making a mistake is even a a positive thing. Yeah, so there's a paper in a little journal called Annals of Mathematics.
好吧，我们真的是投入了太多。所以我们就一直坚持下去。最后我们花了两年时间才解决这个问题。我们最终找到了一条完全不同的路径来解决这个问题。而实际上，这篇论文是我最引以为傲的论文之一。它还得了奖。你知道，如果我们没有犯下这个早期成功带来的错误，我们早在解决这个问题之前就放弃了。所以实际上，有时候犯错甚至是一件好事。是啊，所以有一篇论文发表在一本叫《数学年刊》的小刊物上。
07问蠢问题：斯坦不等式的反例
17:20
He's very happy with it and he's my four co-authors. Um Yeah, so yeah, freedom to fail. There's a corollary to that which is that um you know, in order to make progress in math you have to ask lots and lots of really stupid questions. And this is something which we said you know, um Often we uh we sort of condition our students to not speak up unless they're really confident in their answer and that's actually often is is is the wrong approach. You know, students often ask you know, silly questions in you know, when they're solving the math problems which you know, if you know the answers then why would you ask this question? It's dumb. But actually these these these questions are really important to answer.
他对此非常满意，还有我的四位合作者。嗯，是啊，所以，失败的自由。这里有一个推论，那就是，嗯，你知道，为了在数学上取得进展，你必须问大量大量非常蠢的问题。这是我们说过的一点，嗯，我们常常，呃，我们某种程度上训练学生，除非他们对自己的答案非常有把握，否则就不要发言，而这其实往往是错误的做法。你知道，学生在解数学题的时候经常会问一些，你知道，很傻的问题，你知道，如果你知道答案的话，那你为什么还要问这个问题呢？这很蠢。但实际上，这些问题是非常值得回答的。
18:03
And you know, I mean okay, I've listed some some silly questions here that a student might ask but you know, mathematician like some of the deepest progress in mathematics has has come from mathematicians asking similarly stupid questions just at a slightly higher level. I won't each one of these questions can lead to an hour long talk but I'm I'm not going to to but yeah, there there are You can almost reduce every new breakthrough in mathematics to someone asking a stupid question. Um You know, yeah and Paul Halmos was really what's a very famous for sort of teaching students how to how to think like a mathematician. You know, really emphasized that you know, you you should always not just accept what your your teachers tell you as as as you as a static thing. You know, you you have to really fight it and really make it your own. And and one way is just to ask you know, ask your own your own dumb questions.
你知道，我是说，好吧，我在这里列了一些学生可能会问的傻问题，但你知道，数学家——数学中一些最深刻的进展，正是来自数学家提出类似的蠢问题，只不过层次稍微高一点。我不会——这里的每一个问题都可以引出一场一小时的讲座，但我不打算展开讲，不过是啊，确实——你几乎可以把数学中每一项新突破都归结为某个人问了一个蠢问题。嗯，你知道，是啊，保罗·哈尔莫斯非常有名的一点，就是他教学生如何像数学家一样思考。你知道，他真的很强调，你知道，你不应该只是把老师告诉你的东西当作一成不变的东西接受下来。你知道，你必须真正去质疑它，真正把它变成自己的。而其中一种方式，就是去问，你知道，问你自己的那些蠢问题。
19:04
Um I'll just give you one one of example from personal experience. You know, so um Um Yeah, so my advice I had a my graduate advisor Elias Stein was a accomplished mathematician at Princeton. There was an inequality he had proven in all cases except one at the end point and at some point I said I'm going to prove the end point case of Stein's inequality. This was you know, I I tried both by myself and with with some co-authors. We got some very weak partial results but it was one of my dreams to sort of complete this this result that my advisor had worked on.
嗯，我再讲一个我个人的经历。你知道，嗯，是啊，我的导师——我的博士导师伊莱亚斯·斯坦是普林斯顿一位很有成就的数学家。他证明过一个不等式，除了端点情形之外的所有情况都证出来了，某个时候我就说，我要去证明斯坦不等式的端点情形。这个，你知道，我既自己试过，也和一些合作者一起试过。我们得到了一些非常弱的部分结果，但把我导师做过的这个结果补全，一直是我的梦想之一。
19:40
And at some point I was invited to a I think it was 80th birthday conference and I so I had to present something and I I said well I was working on this and I had some partial results and so I I presented what I had. Um And then a colleague of mine in the audience just asked a question, "Have you ever tried to disprove Is there Is there any reason to expect why this inequality is true? Is there a counterexample?" I had This had never occurred to me that the thing I was trying to prove could actually be false.
有一次我受邀参加一个——我想是他八十岁生日的会议，所以我得讲点什么，于是我就说，好吧，我一直在研究这个，我有一些部分结果，所以我就把手头的东西讲了出来。嗯，然后台下我的一位同事问了一个问题：“你有没有试过反过来证伪它？有没有任何理由让人预期这个不等式是成立的？会不会存在反例？”我从来——我从来没有想到过，我一直试图证明的东西居然可能是错的。
20:06
Uh so, I had no good answer to this question. Um I then spent the evening trying to see if it could be false, and actually within a week I had found a counterexample. Um So, you know, it's it's it's a you know, sometimes you know So, it it's not necessarily you that have to ask a dumb question. Somebody has to ask the dumb question. Um Yeah, so just to summarize, these are These are three aspects of being a mathematician. So, you have to learn both intuition and rigor, but combine them eventually into some sort of post-rigorous mindset. You have to embrace the freedom to fail, and you have to ask dumb questions. Thank you very much.
呃，所以我对这个问题没有什么好的回答。嗯，那天晚上我就开始琢磨它是不是可能不成立，结果不到一周我就找到了一个反例。嗯，所以，你知道，这就是——有时候你知道，所以，未必非得是你自己去问那个蠢问题。总得有人来问这个蠢问题。嗯，是啊，那么总结一下，这就是做一名数学家的三个方面。你必须同时学会直觉和严谨，但最终要把它们结合成某种后严谨的思维方式。你必须拥抱失败的自由，你还必须问蠢问题。非常感谢大家。
20:41
>> [applause]
>> [掌声]
08协作、分工与数学的工业革命
20:50
>> WE HEARD SOME THEMES FROM Terry that are actually being echoed in the field of mathematics education right now. And in fact, over the last decade there have been ongoing efforts in mathematics education to more carefully develop and justify ideas, and to openly acknowledge and support struggle and failure and perseverance in problem-solving in classrooms, and to prioritize process including contrasting and learning from multiple solutions to rich problems. And so, we know that the field of education is trying very hard to develop the habits of mind of professional mathematician in our students in our classrooms.
>> 我们从特里那里听到的一些主题，其实正是当下数学教育领域正在呼应的东西。事实上，在过去十年里，数学教育领域一直在努力，让学生更严谨地发展和论证想法，并且公开地承认和支持课堂上解题过程中的挣扎、失败与坚持，同时重视过程，包括对比并从丰富问题的多种解法中学习。所以，我们知道，教育领域正在非常努力地在课堂上培养学生具备专业数学家的思维习惯。
21:34
And so, um I thought I would start off the discussion with a question of my own for you, Terry. >> Okay. Um and I wanted to ask uh some other themes in math education right now are around the ideas and the role of collaboration between students in developing an understanding of the mathematics they're learning, and also the role of technology in the classroom. And I wondered what if you would be willing to share with us some of your thoughts on the role of collaboration and the use of technology in the professional work of a mathematician. Those are great topics.
所以，嗯，我想先用我自己的一个问题来开始这场讨论，特里。>> 好的。嗯，我想问一下，呃，目前数学教育中的另一些主题，是围绕学生之间协作在理解所学数学中的作用和意义，以及技术在课堂中的作用。我在想，你是否愿意和我们分享一下你对协作的作用，以及技术在数学家专业工作中的运用的一些看法。这些都是很好的话题。
22:08
Yeah, so that's those those list of three things I I gave my talk that should not be viewed as a complete list of all the things you should learn as a mathematician. Oh, sorry. No, it's okay. >> Um Yeah, so uh one skill that has become increasingly important over time in mathematics is is the soft skill of knowing how to collaborate. Um I think math has become has evolved from being a very sort of solitary activity uh to to a very collaborative one. Uh partly because the the nature of the subject has has has changed. We work with much more interdisciplinary complex problems that no one person can solve.
是啊，所以我在演讲中给出的那三点，不应该被看作是你作为数学家需要学习的所有东西的完整清单。哦，抱歉。不，没关系。>> 嗯，是啊，呃，有一项技能在数学中随着时间推移变得越来越重要，那就是懂得如何协作这项软技能。嗯，我认为数学已经从一种非常孤独的活动，演变成了一种非常协作的活动。呃，一部分原因是这个学科的性质发生了变化。我们处理的是跨学科得多、复杂得多的问题，没有哪一个人能独自解决。
22:43
Um but also new technology like the internet allows us to to work together on a scale that we cannot do before. Um Yeah, so um I teaching um students to to to work together on on on group projects is an important um skill. So, we um which is a It's it's it's It's a very different type of skill. So, yeah, it's a you know, people skills are becoming more and more important, which traditionally we've not not been a strong suit, to be honest, in mathematics, but uh we are evolving. Um And yeah, we'll be collaborating not just with with other humans, but with with increasingly with with AI. Um Yeah, they are AI tools are becoming very powerful, in some ways still very weak in others, um but when you collaborate, the the the beauty of collaboration is division of labor, you know. So, um in the past, a mathematician had to handle every single aspect of of a problem-solving process, you know, identify the problem, identify the strategy, identify a good strategy, um uh implement it, and and then write it
嗯，但同时，像互联网这样的新技术，也让我们能够以以前做不到的规模一起工作。嗯，是啊，所以，嗯，教学生，嗯，如何在小组项目中合作，是一项重要的，嗯，技能。所以，我们嗯，这是一种——这是一种非常不同类型的技能。所以，是啊，你知道，人际交往能力正变得越来越重要，而说实话，这在数学界传统上并不是我们的强项，但呃，我们正在演变。嗯，而且，是啊，我们将不只是与其他人协作，还会越来越多地与人工智能协作。嗯，是啊，AI 工具正变得非常强大，但在某些方面仍然非常弱，嗯，不过当你协作时，协作的美妙之处就在于分工，你知道。所以，嗯，在过去，一个数学家必须处理解题过程的每一个环节，你知道，识别问题、确定策略、找到一个好的策略，嗯，呃，实施它，然后把它写
23:46
up, explain it. Um And and now with you you can collaborate with other humans and and AIs and and and different members of the collaboration will do different tasks. Um So, yeah, we are we are slowly moving into the where we have we experiencing some sort of industrial revolution. Um and I our class our teaching will have to evolve. Uh It is changing very fast for better or for worse. But I think, you know, we have good resources like like this, we we can adapt. Thank you so much. I noticed that there was a question from the Zoom room.
出来、解释清楚。嗯，而现在，你可以和其他人协作，也可以和 AI 协作，协作中的不同成员会承担不同的任务。嗯，所以，是啊，我们正在慢慢进入——我们正在经历某种工业革命。嗯，而且我们的课堂、我们的教学也将不得不随之演变。呃，它变化得非常快，不管是好是坏。但我认为，你知道，我们有像这样的好资源，我们是能够适应的。非常感谢。我注意到 Zoom 会议室里有一个问题。
09大语言模型的真实成功率
24:24
>> Yes, I was a couple of your panelists who mentioned real-world applications using mathematics. And I was thinking one of the best real-world applications is if you could get a piece of the UCLA endowment fund, rumored to be worth $5 billion or so, and take the incredible brainpower of the mathematics department, I don't know if you want to work with the finance department, but the average 10-year return from the fund is under 8%. And I'm sure with using AI or numerical simulations or uh what what whatever you guys choose to do, you could you could crush that average return.
>> 是的，你们有几位嘉宾提到了数学在现实世界中的应用。我在想，最好的现实应用之一，就是如果你们能拿到 UCLA 捐赠基金的一部分——传闻大概价值 50 亿美元左右——然后动用数学系那些惊人的智力，我不知道你们是否愿意和金融系合作，但这个基金的十年平均回报率还不到8%。我相信，用 AI 或者数值模拟，或者呃，你们选择的任何方法，你们都能大幅超越那个平均回报率。
25:08
And uh >> [laughter] >> So, you know how I said in my my my presentation that in mathematics failure is very cheap. Um so, uh there's an asterisk there. If you if you connect it to the real world, you do actually have to do some serious risk-benefit analysis. I think teaching financial modeling would be would be a great school project, actually. Um so, um and and yeah, you could you could run sort of some some simulated markets and and see how just a very small bit. I wouldn't directly send the students to to manage the UCLA endowment just yet.
还有呃 >> [笑声] >> 所以，你知道我在演讲中说过，在数学里失败是非常廉价的。嗯，所以，呃，这里有个附注。如果你把它和现实世界联系起来，你确实必须做一些认真的风险收益分析。我觉得教金融建模会是一个很棒的学校项目，真的。嗯，所以，嗯，而且是啊，你可以，你可以运行一些模拟市场，然后看看，只是很小规模地试一下。我暂时还不会直接派学生去管理 UCLA 的捐赠基金。
25:49
>> [laughter] >> So, we have another question in the room. Would you manage my portfolio? Is that >> [laughter] >> Yeah, I refer to my previous comment. >> [laughter] >> I I do have a serious question. How good is How good are large language models now at doing advanced theoretical math? Like, can they push the boundaries forward in the field? They are becoming increasingly capable at at many things. Um so, I think at this point they're on the level of kind of like It's like having an army of graduate students of various levels of quality, and you you you give a problem, and they will try they'll dig up in the literature some techniques and try them one by one. Sometimes they they succeed, sometimes they fail. Uh so, sometimes they they do spectacularly well, uh sometimes they they fall flat on their face.
>> [笑声] >> 好，现场还有另一个问题。你愿意帮我管理投资组合吗？这算 >> [笑声] >> 是啊，我引用我刚才那句话。>> [笑声] >> 我确实有个正经问题。现在的大语言模型在做高等理论数学方面有多厉害？比如说，它们能推动这个领域的前沿吗？它们在很多方面正变得越来越有能力。嗯，所以我觉得目前它们大致相当于拥有一支水平参差不齐的研究生大军，你给它们一个问题，它们会试着从文献里翻出一些技巧，一个一个地试。有时候它们会成功，有时候会失败。呃，所以，有时候它们表现得非常出色，呃，有时候它们会栽个大跟头。
26:41
Um So, if you only if you only are going on social media Hm? Not quite sure what's happening here. If you If you only go on social media and and you see people talk their the the biggest successes, it looks amazing. Um and then you try to download these tools or or use them use them at home on your favorite problem, and the success rate is often like 1 to 2%. Um So, um I think there's certain types of math where they'll be very useful. Um I think if there's a problem that you can solve kind of a task that you can split up into like 100 subproblems, and um and um you don't mind if there's a tool that can only solve 10% of them.
嗯，所以，如果你只是……如果你只是上社交媒体——嗯？不太确定这里出了什么状况。如果你……如果你只上社交媒体，看到大家谈论他们最大的成功案例，那看起来简直太棒了。嗯，然后你去下载这些工具，或者用它们……在家里用它们来解决你最关心的问题，成功率往往只有 1% 到 2%。嗯所以，嗯，我觉得在某些类型的数学里它们会非常有用。嗯，我觉得如果有一个你能解决的问题，或者说一个任务，你可以把它拆成大概 100 个子问题，嗯，而且，嗯，你不介意有一个工具只能解决其中 10%。
27:20
That's 10 problems you got to solve, and then another tool will solve another 5%. Um so, I I I think we as I said, we math has to industrialize and start playing the games of percentages. So, instead of working on taking one problem and working on it very very hard, which is that our usual MO, you know, um creating these large-scale projects where we just accept that there's a partial success rate. Um then these tools will become quite quite powerful, I think. But it's it's a very fluid situation. It's it's exciting, but also a bit scary, too. But uh Yeah, um I think there's more good than bad.
那就是你搞定了 10 个问题，然后另一个工具再解决另外 5%。嗯，所以我……我想，就像我说的，我们……数学必须工业化，开始玩百分比的游戏。所以，与其盯着一个问题非常非常努力地钻研——那是我们通常的做法，你知道的，嗯——不如去创建这些大规模的项目，我们只需接受它有一个部分成功率。嗯，那样这些工具就会变得相当相当强大，我是这么认为的。但这是一个非常流动的局面。这很令人兴奋，但也有点吓人。不过呃，是的，嗯，我觉得好处多于坏处。
10神童教育与犯错的特权
27:59
I am so curious about when you were introduced, and then um Mr. Rope or Professor Roper, sorry, talked about once you become a parent, what you see your children experience in school. Um and then I immediately went to gosh, at 10 years of age winning a math competition, um you're, you know, a a prodigy, a celebrity, you know, kind of like the smartest one. Everyone knows in school who the smartest kid is. It's not easy. Fast forward to today, you can stand up here and say as an adult, as a parent, there are no dumb questions. Um I still think I ask really dumb questions, and I'm terrified to do it publicly.
我特别好奇，在介绍你的时候，然后，嗯，Rope 先生……抱歉，是 Roper 教授，谈到了当你成为父母之后，你会看到自己的孩子在学校里经历些什么。嗯，然后我立刻就想到，天哪，10 岁就赢得数学竞赛，嗯，你就是，你知道的，一个神童、一个名人，你知道，有点像是最聪明的那个。学校里每个人都知道谁是最聪明的孩子。这并不容易。快进到今天，你能站在这里，以一个成年人、一个父母的身份说，没有愚蠢的问题。嗯，我到现在还是觉得自己会问非常蠢的问题，而且我很怕当众问。
28:48
Um and I'm doing it right now, cuz I feel like I'm probably asking some But um the point being is where you are today, um looking at the future of where you talk about AI, LLMs, the future of math, the future of uh education, um the system, and so it's how wrong wrong it is, um what would you say to the schools, the teachers, the districts? How you bring math into the world that keeps kids wanting to work hard to ask dumb questions, cuz I can't imagine how somewhat easy it might have been for you, but at the same time how hard um I can't imagine what you went through.
嗯，而我现在正在这么做，因为我觉得我大概正在问一些……但重点是，你今天所处的位置，嗯，展望未来，你谈到 AI、大语言模型、数学的未来、教育的未来，嗯，这个体系，以及它到底有多不对劲——嗯，你会对学校、老师、学区说些什么？该怎么把数学带进这个世界，让孩子们始终愿意努力、愿意问蠢问题？因为我无法想象对你来说这可能有多轻松，但同时又有多难，嗯，我无法想象你经历了什么。
29:34
Is that a fair question? >> Yeah, well, um yeah, it was a different time. Um I think um yeah, so the world was a lot slower then, um and more predictable. Um and actually um so, I had a very accelerated education actually in skipping five grades. Um and local headmasters at the high school and and elementary school and and at the local university had to design a special curriculum just for me actually. And somehow it was all very new and they weren't It's so many rules in place that I mean there wasn't a structure structured gifted education program at the time. Everyone was improvising.
这个问题合理吗？>> 是的，嗯，是的，那是一个不同的年代。嗯，我想，嗯，是的，那时的世界慢得多，嗯，也更可预测。嗯，而且其实，嗯，我接受的其实是非常超前的教育，跳了五级。嗯，当地高中和小学的校长，还有当地大学的人，不得不专门为我设计一套特别的课程。不知怎么的，这一切都非常新鲜，他们也没有……当时并没有那么多规矩，我是说，并没有一个既定的结构当时并没有结构化的资优教育项目。大家都是临场发挥。
30:13
And it kind of worked out through through all the efforts of of my parents and and my um and all all the various administrators. Um So in in many it's in many ways it's better and worse. Now I mean so in my So this is in my home country of Australia. Now they have gifted programs which are much more structured. So you don't have to improvise everything from scratch, but they're also maybe a bit more rigid. Some some things that I I could get away with I could could would not be be possible now. So it's a different world. Um Uh so I mean I don't know how what works and what doesn't, but you know we have this amazing center here that is devoted to that.
但靠着我父母、还有我……还有各位管理者的努力，最后也算是走通了。嗯，所以在很多方面，现在既更好也更糟。我是说，在我的……这是在我的祖国澳大利亚。现在他们有了资优项目，结构要完善得多。所以你不必事事从零开始摸索，但它们可能也更死板一些。有些事情我当年能蒙混过关，现在就不可能了。所以这是个不一样的世界。嗯，我是说，我不知道什么管用、什么不管用，但你知道，我们这里有一个了不起的中心，就是专门做这件事的。专门做这个。
30:58
So you know I mean we have to try experiments and showcase the successes also learn from our failures. Um and uh um you know I'm sure we can do better than than the way we do now. And you know um one thing we learn as a mathematician is even a partial solution is better than no solution at all. So um so I I and I think that's that's what the center is all about. Thank you Terry. Um what you just said reminded me of something I said to Esei nine months ago or so. I said let's not be afraid of failing, let's be afraid of not learning.
所以你知道，我是说，我们必须去做实验，展示成功的经验，也从失败中学习。嗯，还有……嗯，你知道，我相信我们能做得比现在更好。还有，你知道，作为数学家我们学到的一点是：哪怕只有部分解，也好过完全没有解。所以，嗯，我觉得这正是这个中心的意义所在。谢谢你，Terry。嗯，你刚才说的让我想起大概九个月前我对 Esei 说过的一句话。我说：我们不要害怕失败，要害怕的是没有学到东西。
31:36
So Um all right so Kate Stevenson here professor at CSUN. Hi there. So I've just up the road. I teach the dread proofs class at Cal State Northridge and I do it on purpose as chair of math because it gives me access to my majors. It's a wonderful transition that happens to the students and I love watching it happen. >> It's one way to put it. >> I could get you on video telling them that it's okay to fail uh that would help my job so much. But that is I think the piece that you describe these three pieces of what it means to be a mathematician what is maybe missing from that framework is when you're talking to students who weren't in math camp, who didn't succeed in high school. I have first generation students, the majority of my students. I have students who've had very difficult backgrounds and current existence.
那么，嗯，好的，我是 Kate Stevenson，CSUN（加州州立大学北岭分校）的教授。你好。我就在这条路往前一点的地方。我在加州州立北岭分校教那门让人闻风丧胆的证明课，而且我是故意去教的，因为身为数学系主任，这让我有机会接触到本系的学生。学生身上会发生一种美妙的转变，我很喜欢看着它发生。>> 这是一种说法。>> 我真该把你说“失败没关系”的话录下来，那对我的工作会有莫大帮助。但我想，你描述的成为数学家的这三个要素里，这个框架里可能缺失的一块是：当你面对的学生没上过数学夏令营、高中时也不算成功。我的学生里有第一代大学生，而且占多数。我有些学生的成长背景非常艰难，当下的处境也很艰难。
32:34
Building the trust with those students to ensure that they are willing to make the mistakes because it is a privilege to be able to make mistakes and feel safe doing that. And I think one of the ways we can collaborate towards building that trust is speaking with one voice and recent about the fact that mathematics is about making mistakes, persisting, that mathematics is a web of connected ideas, not a bunch of algorithms. And happily recently the CSU, the UCs, and the community colleges have come together on a statement of what it means to be prepared for post secondary study and it reflects all of the things that you spoke of. So I think lifting those voices up to say we're not arguing about this, we agree on this point.
要和这些学生建立信任，让他们愿意去犯错，因为能够犯错、并且在犯错时感到安全，本身就是一种特权。我觉得，我们可以携手建立这种信任的方式之一，就是用同一个声音说话，就近来说，要讲清楚：数学就是关于犯错、坚持下去的，数学是一张相互连接的思想之网，而不是一堆算法。可喜的是，最近 CSU、UC 系统和社区学院已经共同发表了一份声明，界定什么叫做为高等教育做好了准备，而它反映了你所说的所有这些内容。所以我认为，要把这些声音提升起来，告诉大家：我们在这件事上没有分歧，我们是有共识的点。
11AI 能否创造定义与阶段是否必然
33:25
Thank you Kate. We have an additional question from the Zoom room and it's from a former student of Phil's named Chris Ennis who had to actually leave the Zoom room, but I did want to make sure that his question was asked. Um so this is a question for you Terry. Have you ever seen an instance of an AI creating a new mathematical definition in order to prove a result or solve a problem? I think coming up with good definitions is one of the most creative activities mathematicians do. For example, the definition of compactness in topology makes proving a lot of fundamental theorems quite a bit easier. Do AIs do that?
���谢你，Kate。我们在 Zoom 会场还有一个问题，来自 Phil 以前的一位学生，叫 Chris Ennis，他其实已经中途离开 Zoom 了，但我还是想确保他的问题被提出来。嗯，所以这个问题是问 Terry 的。你有没有见过这样的例子：AI 为了证明某个结果或解决某个问题，创造出一个新的数学定义？我觉得想出好的定义是数学家最有创造性的工作之一。比如说，拓扑学中紧致性的定义，让很多基本定理的证明变得容易许多。AI 会做这种事吗？
34:10
Um yes and no. So you can make an AI generate all kinds of new ideas and definitions and but it would maybe generate one good definition and like nine bad ones. And it doesn't You can't really tell which which ones are good which ones are bad without really um going through it yourself. Um And also some parts of math are very checkable. You know like if you either you solve your problem or you don't. And the parts of math that are very verifiable AIs are actually pretty useful currently. But with things like a definition you can't you can't really score you know out of 10 how good a definition this is.
嗯，算是会，也算不会。你可以让 AI 生成各种各样的新想法和新定义，但它可能生成一个好定义，同时冒出九个糟糕的。而且你没法真正分辨哪些好、哪些差，除非你自己把它们认真过一遍。嗯，另外，数学的某些部分是非常可检验的。你知道的，就是你要么解出这个问题，要么没解出来。而在那些非常容易验证的数学领域，AI 目前其实还挺有用的。但像定义这种东西，你没法真的给它打个分，说这个定义能打十分里的几分。
34:51
And so so those type of tasks these AIs are still very frustrating. They may occasionally hit gold, but then they just pass they just move on and then then the the next nine things they say are rubbish. So I haven't seen that yet. Maybe the technology will improve, but yeah not yet is basically my answer. So you you mentioned three phases of education. There's the pre-rigorous, the rigorous, and the post-rigorous. And my question is is that how it should be? And if not, are there ways to expose people who haven't been exposed to those later stages to maybe taste it or get an understanding of it without getting that full education formal education experience? That's that's a good question.
所以对这类任务，这些 AI 还是很让人抓狂。它们偶尔可能撞上金子，但接着就一带而过，直接往下走了，然后接下来说的九件事全是废话。所以我还没见过那种情况。也许技术会进步，但基本上我的回答是：还没有。你提到了教育的三个阶段：前严格阶段、严格阶段和后严格阶段。我的问题是，事情本来就该是这样吗？如果不是，有没有办法让那些还没接触到后面这些阶段的人，也能尝一尝那种滋味，或者有所领会，而不必经历完整的正规教育？这问题问得好。
35:36
Um I know if you rush it um there are people who are like There are some people who are like really really enthusiastic about math and they have always great ideas but they have no rigor. And they try to jump to the post-rigorous stage before they they they know how to prove things. And uh it's it it can be a little bit awkward because a lot of what they say doesn't actually make sense, but you can't even they don't even understand why. Um I think it's important to have people from different stages of this education process talk to each other more.
嗯，我知道如果你操之过急……有些人是那种，有些人对数学真的非常非常热情，脑子里总有很棒的想法，但完全没有严格性。他们想在还不会证明东西之前，就直接跳到后严格阶段。呃，这会有点尴尬，因为他们说的很多东西其实讲不通，而你甚至没法……他们自己都不明白为什么讲不通。嗯，我觉得很重要的一点是，让处在这个教育过程不同阶段的人多互相交流。
36:09
Um and just to know that that that what you're experiencing now is not the final state of mathematics. Um but just um so even if you're not ready to appreciate sort of the full uh aspects of it just take just get get a taste. You know again we we don't show our process nearly as much. Um you know um there there are other activities where people love just watching. You know it's like like computer games there's a whole culture of people watching other people play computer games or watching athletes train or watching actors do their outtakes. Um we we don't have the same culture in mathematics of we we think what we do is either very private or very boring.
嗯，也让大家知道，你现在所经历的并不是数学的最终形态。嗯，不过就算你还没准备好去欣赏它完整的那些方面，也可以先尝个味道。你知道，还是那句话，我们展示自己过程的次数太少了。嗯，你知道，还有一些活动，人们光是看着就很享受。比如说电子游戏，有一整套文化是人们看别人打游戏，或者看运动员训练，或者看演员的 NG 片段。嗯，我们在数学里就没有这种文化，我们觉得自己做的事要么很私密，要么很无聊。
36:55
But maybe if we shared more of what we did then this would help. Perhaps that's why that YouTube video that you recently posted has 100,000 views because you're in that video So there's a video that Terry shared with me in which he's using actively using an AI to engage in research and it's gained quite a number of views and maybe that's a little bit of taking the lid off and showing what's under the hood in real time. Yeah activity. Yeah we don't do it nearly enough. I I wanted to say that uh you're mentioning how failure is an important part what we learn from our mistakes. And this brought back to for me um this a study um of the third international math and science study that James Stigler did uh where he went to eighth grade classrooms and videotaped in Japan, Germany, and the US.
但也许如果我们多分享一些自己做的事情，就会有帮助。也许这就是为什么你最近发的那个 YouTube 视频有十万次观看，因为你出现在那个视频里。Terry 分享给我一个视频，里面他正在实际使用 AI 来做研究，这个视频获得了相当多的观看量，也许这就有点像是揭开盖子，实时展示引擎盖底下是什么样的。是啊，这种活动。是的，我们做得远远不够。我想说的是，呃，你提到失败是很重要的一部分，我们从错误中学习。这让我想起了，嗯，James Stigler 做的第三次国际数学与科学研究，他去了日本、德国和美国的八年级课堂做录像。
38:06
And one of the things that we noticed was that in the American classrooms if a kid put his problems on the board and he made a mistake ooh you know the it was not seen as a good thing. However, in Japan when a kid put his work on the board and there was a mistake the teacher made a big thing out of how grateful the class was because they could learn from that mistake. And so the whole difference in pedagogic outlook on praising what we can learn from what a mistake was was exactly what I got from what you were saying today.
我们注意到的一点是，在美国的课堂上，如果一个孩子把自己的解题过程写在黑板上，结果出了错，哎呀，你知道的，那不会被看作是好事。然而在日本，当一个孩子把解题过程写在黑板上、出现了错误时，老师会大力强调全班有多么感激，因为大家可以从这个错误中学习。所以在教学观念上，对于'我们能从错误中学到什么'这种赞赏的差别，正是我今天从你所说的话里体会到的。
12卡在同一扇门前：做数学的感觉
38:57
I'm going to ask a very Southern California question. Like how does it feel to do math? Like what is the feeling when it goes well? And what's the feeling when it goes I don't know. We're we're not we're no longer going to use the word wrong, right? We're going to say like productively productively failing? Well, so when I was a kid I played way too many computer games. It was one of my favorite hobbies. Um and back in the '90s uh when the internet was not so widespread, if there was a computer game and you could not solve it, you know you could not find the key to open this door there was no internet walk-through.
我要问一个非常南加州风格的问题。做数学是什么感觉？顺利的时候是什么感觉？不顺的时候又是什么感觉——我也不知道该怎么说，我们不再用'错'这个词了，对吧？我们要说'富有成效地失败'？嗯，我小时候玩电子游戏玩得太多了。那是我最喜欢的爱好之一。嗯，在九十年代，那时候互联网还不太普及，如果有个游戏你过不去，你知道，你找不到打开那扇门的钥匙，那时候是没有网上攻略的。
39:36
Maybe like there was the your local game store would sell some some some solution book or something, but basically you only things you every night every night you play the game you get stuck at the same damn door, okay, and you you uh just over and over again. Um but then, you know, one day you just have this brainwave and you do kind of this rock and up there's a key, okay, and like you you uh now you can get through. Um and that was really satisfying. Um and um like it feel it it felt earned um that um you you work at a problem and you you you explore it explore it. Um and so sort of uh you know, even though you don't look like you're making progress, often what you're doing is that you're removing the negative space of all the wastes of the problem that don't work until there's only one path left. Um and then you you can and and then you're ready to to to find that state.
也许你家附近的游戏店会卖某种攻略书之类的，但基本上你能做的就是每天晚上，每天晚上你玩这个游戏，都卡在同一扇该死的门前，好吧，一遍又一遍。嗯，但是后来有一天，你突然灵光一闪，去搬开某块石头，底下就有把钥匙，好了，然后你就能过去了。嗯，那真的太有成就感了。嗯，而且那种感觉像是自己挣来的，你钻研一个问题，不断地探索、探索。嗯，所以某种意义上，你知道，虽然看起来你没什么进展，但你实际上在做的，是把问题里所有走不通的死路——那些负空间——一点点排除掉，直到只剩下一条路。嗯，然后你就能……然后你就准备好抵达那个状态了。
40:26
Um so um and then like often once you get it, it's like, "Oh, how come I didn't see that before?" Um and the reason you didn't see it before is cuz you didn't put in the work to really sort of clear out all all all the extraneous rubbish. Um So, um okay, I I I won't do a kids these days thing, but you know, I mean um I mean, you know, technology has improved our lives in many ways. Uh but it's it's given us instant access to all kinds of things and but by the same token, we expect instantaneous solutions for almost anything.
嗯，所以……然后往往一旦你想通了，就会觉得："哦，我之前怎么没看出来呢？"嗯，你之前没看出来的原因是，你还没下够功夫去真正把所有那些无关的杂乱东西清理干净。嗯，所以……好吧，我不会说"现在的小孩啊"这种话，但你知道，我是说，科技在很多方面改善了我们的生活。呃，但它也让我们能即时获取各种东西，可与此同时，我们几乎对任何事情都期待即时的解决方案。
40:57
Um but then when it comes to maths, uh it's one of the few places left where uh you don't get instant answers. Well, actually now we're paying but um you know, but when you're working without AI assistance, you do not get uh these instant answers um and you have to work at it and um but when you do get the solution, it it does feel earned and that that that is that is a nice feeling. You know your 2-year uh problem where you got it wrong after a couple weeks it was going to publish. If AI if you applied AI today to that, what how long would that take? Uh I have not tried. I'm a little bit a little bit uh I'm a bit scared to. But uh Well, for better or for worse, um the way these AI are trained is that they they they suck up every single um um thing on the internet that they can and they train on it. So, this paper that we published 10 years ago, it's in the training data. Um so um either consciously or unconsciously, I'm putting quotes, if you ask an AI this question, um they are very very good at
嗯，但说到数学，呃，这是所剩不多的、你得不到即时答案的领域之一。嗯，当然现在我们也在付出代价，但你知道，当你在没有 AI 辅助的情况下工作时，你是得不到这些即时答案的，你必须下功夫去钻研。嗯，但当你真的得到解答时，那种感觉是靠自己挣来的，那……那是一种很好的感觉。你说到你那个做了两年的问题，几周后发现做错了，本来是要发表的。如果今天你把 AI 用在那个问题上，要花多长时间？呃，我还没试过。我有点……有点……呃，我有点不敢试。不过呃，不管是好是坏，嗯，这些 AI 的训练方式就是把互联网上能找到的每一样东西都吸进去，然后拿来训练。所以，我们十年前发表的那篇论文，就在训练数据里。嗯，所以，不管是有意识还是无意识地，我要打个引号，如果你拿这个问题去问 AI，嗯，它们非常非常擅长复现互联网上
41:55
reproducing things that have been done somewhere on the internet and um we can't tell for sure. So, I'm sure they'll do well, but whether it's because they memorized it or they thought came up with originally, it's no one really knows. What if the problem was in the original proof? Um they've been able to come up with they've solved some problems that haven't been solved before. Uh you look at how they do it did it and it's often like a remix of a method that worked for a related problem. Um I haven't seen a something that really stunned me like no one has ever seen this proof before, but it's it's very very useful.
某个地方已经做过的东西，而且我们没法确定。所以我相信它们会做得不错，但这究竟是因为它们记住了，还是它们自己原创想出来的，谁也说不清。那如果问题出在原来的证明里呢？嗯，它们已经能够……它们解决了一些以前没被解决过的问题。呃，你去看它们是怎么做到的，往往像是把一个用在相关问题上的方法重新混搭了一下。嗯，我还没见过那种真正让我震撼的、从来没有人见过的证明，但它确实非常非常有用。
42:34
Um whether it's very creative, that's still under debate. All right, thank you for Terry so much for fielding all of those questions. We appreciate your time. >> [applause]
嗯，至于是不是很有创造力，这一点还有争议。好，非常感谢 Terry 回答了所有这些问题。感谢你抽出时间。>> [掌声]
42:53
>> So I don't talk.
>> 那我就不多说了。
视频总结 · 一句话概括与核心要点
一句话概括
陶哲轩在 UCLA Curtis Center 的演讲中提出"像数学家一样思考"的三大要素——融合直觉与严格的"后严格"思维、拥抱失败的自由、敢于提"蠢问题"——并在问答中讨论了协作、AI 在数学研究中的真实能力与局限。
核心要点
- 数学思维分三个阶段，而大多数人止步于第二阶段：前严格阶段（K-12/K-14，靠例子、直觉、公式计算，允许模糊和出错）；严格阶段（大学证明课，只有一种正确做法，旧直觉被贬为"小孩子的东西"）；后严格阶段（研究生阶段，重新回到直觉与非正式推理，但随时能把"挥手"论证转化为严格证明）。陶认为第三阶段才是数学"好玩"的部分，可惜绝大多数人从未见过。
- 反证法在三个阶段呈现完全不同的面貌：小学生在课间玩"谁能说出最大的数"，最终发现"任何数都可以加一"，实际上已自发完成了"最大数不存在"的反证——但无法用语言表述（前严格）；到了课堂上学 √2 无理性的证明，学生觉得这套"假设成立→推出矛盾"的套路陌生怪异，无法与自己的经验连接（严格）；而职业数学家把反证法当作"一步棋"随手使用，正如 G. H. Hardy 所言，棋手只牺牲一个象或车换取局面优势，数学家却可以"把整盘棋押上去"仍然获胜（后严格）。
- 数学是失败最廉价的学科，但教学方式恰恰相反：工程师设计桥梁出错代价高昂，心脏外科医生切错部位是灾难，而数学证明不成立只需重做。V. Arnold 称数学是"实验便宜的那部分物理学"。研究生训练的核心之一就是不断尝试可能失败的方法，因为失败的方式本身极有价值。陶指出评价标准存在错位：答案必须正确，但过程可以充满失败——两者都要教。
- "分析瘫痪"的破解只需一句"允许失败"：一名学生从三本书学到三种 Taylor 近似公式，不知选哪个而完全卡住。陶只说了一句"先随便试一个，不行再看下一步"，学生就被解锁了。Niels Bohr 的名言呼应此点：专家就是在一个狭窄领域里犯过所有可能错误的人——错误现在不犯，将来也要犯。
- 陶自己的"错误反而成就了获奖论文"：他与四位合作者攻克 Jean Bourgain 未完成的偏微分方程问题，自以为几个月就搞定，甚至已在订餐厅准备开香槟；写稿时一位更细心的合作者发现展开的 13 项中只控制了 12 项，漏掉的第 13 项恰恰是最坏的一项，怎么都无法处理。此时已投入六个月，团队因沉没成本继续硬撑，最终花两年找到完全不同的方法，论文发表于 *Annals of Mathematics* 并获奖。陶坦言：如果没有早期那次"错误的成功"，他们早就放弃了。
- 数学文化只发表胜利，导致普遍的冒名顶替感：读 Fields 奖得主如 Maryam Mirzakhani 的论文全是成功，对照自己证明总是不通的日常，就会觉得自己是骗子。而"迷路"其实是数学家的默认状态，陶主张更多公开失败过程，把它正常化。
- 几乎每个数学突破都源于有人提了一个"蠢问题"：Paul Halmos 强调不要静态接受老师所教，要与之搏斗、据为己有。陶的亲身经历：他多年试图证明导师 Elias Stein 不等式的端点情形，只得到弱的部分结果；在一次 80 岁纪念会议上报告时，一位同事问"你试过反证它吗？有理由相信它是对的吗？"——陶从未想过这可能是假的，当晚开始找反例，一周内就找到了。提蠢问题的人不一定是你自己，但总得有人提。
- AI 目前像"一支水平参差的研究生军团"，成功率约 1–2%：它们会翻文献逐一尝试技巧，有时惊艳有时惨败；社交媒体只展示成功案例，自己拿去用时成功率极低。陶认为数学必须"工业化"、玩"百分比游戏"：把大任务拆成 100 个子问题，接受 A 工具解决 10%、B 工具再解决 5% 的模式，而非传统的一人死磕一题。AI 能生成新定义，但"一个好的配九个烂的"，且定义好坏不像解题那样可验证打分，因此在需要创造性判断的任务上仍令人沮丧；已解决的新问题多是"相关方法的混搭"，尚未见到真正让人惊叹的全新证明。
- 协作已成为数学家越来越核心的软技能：数学从孤独活动演变为高度协作，原因是问题越来越跨学科、复杂到无人能独立完成，加上互联网使大规模合作成为可能。协作的价值在于分工——过去一个数学家要包揽识别问题、选策略、执行、写作全流程，未来将与人类和 AI 分担不同环节，数学正经历某种"工业革命"。
结论与值得注意的细节
- 陶总结的三条：兼学直觉与严格并最终融合为后严格思维；拥抱失败的自由；提蠢问题。他特别指出这不是完整清单，协作能力同样重要。
- 关于"能否跳过阶段直达后严格"：陶警告有些人热情高涨、点子多但毫无严格性，试图在学会证明前就跳到后严格阶段，结果说的话大多不成立而自己却意识不到。他建议不同阶段的人多交流，让学生知道"你现在经历的不是数学的最终形态"。
- 数学家应像游戏直播、运动员训练、演员花絮那样展示"过程"：陶近期一段用 AI 做研究的实录视频获 10 万观看，正是"揭开盖子"的尝试。
- 陶把做数学的感觉比作 90 年代没有攻略的电脑游戏：每晚卡在同一扇门前，某天灵光一现搬开石头找到钥匙——那种成就感是"挣来的"。看似无进展的时间其实是在清除所有走不通的路，直到只剩一条。
- 关于 UCLA 50 亿美元捐赠基金年均回报不足 8% 的玩笑提问，陶回应："数学中失败廉价"有个星号——一旦连接现实世界就需要严肃的风险收益分析，他不会让学生直接管理捐赠基金。
- 对"用 AI 重做那个两年问题要多久"，陶承认不敢试：十年前的论文已在训练数据中，AI 能复现，但无法分辨是记忆还是原创。
- 现场教育者补充：James Stigler 的 TIMSS 跨国课堂录像研究显示，美国课堂上学生板演出错被视为负面，而日本教师会大力表扬全班可以从这个错误中学习——这正是陶所呼吁的教学取向。CSUN 数学系主任 Kate Stevenson 强调，对第一代大学生等弱势背景学生而言，"能安全地犯错"本身是一种特权，需要先建立信任；加州 CSU、UC 与社区学院已联合发布关于大学预备的声明，与陶的观点一致。
- 陶回忆自己跳了五个年级，当时澳大利亚没有结构化资优教育，全靠校长、大学和父母即兴设计课程；如今制度更完善但也更僵化——"部分解总好过没有解"。
核心句型 · 9
1. I like to divide X into what I call A, B, and C
“I like to divide math into what I call the pre-rigorous, rigorous, and rigorous stages.”
提出自创分类框架的标准句式。「what I call」暗示术语是自己命名的，既谦逊又确立主张。适合演讲或论文中引入个人理论模型。
2. If you teach that X, and you don't feel like X, then you don't feel like Y
“If you teach that mathematicians are happy geniuses, then and you don't feel like a genius, then you don't feel like a mathematician.”
条件链推理句式，用两层 if…then 揭示错误叙事的连锁后果。适合论证某种观念如何间接伤害受众。
3. Compared to other disciplines, X is very, very cheap/expensive
“Compared to other disciplines math is actually it's very failure is very cheap.”
通过跨领域对比凸显本领域特点。后接具体例证（工程师、外科医生）使抽象判断具象化。仿写时可换成任何需要对比的属性。
4. It doesn't matter if the first thing you pick doesn't work. Just try one.
“It doesn't matter if the first thing you pick is it doesn't work. Just try one.”
给予「行动许可」的口语句式：先消除对结果的顾虑（It doesn't matter if…），再用祈使句推动。适用于辅导、带团队时解除他人的犹豫。
5. It had never occurred to me that X could actually be Y
“This had never occurred to me that the thing I was trying to prove could actually be false.”
表达「盲点被揭穿」的经典句式，过去完成时强调此前从未想过。could actually 加强反转意味。适合叙述认知转折的时刻。
6. It's not necessarily you that have to do X. Somebody has to do X.
“It's not necessarily you that have to ask a dumb question. Somebody has to ask the dumb question.”
强调句 It's not… that… 先否定个人责任，再用 Somebody 把责任泛化为群体机制。适合论述集体协作中的角色分工。
7. It's like having an army of X of various levels of quality
“It's like having an army of graduate students of various levels of quality”
用「一支……大军」比喻数量多但质量参差的资源。of various levels of quality 是精准的限定，避免过度乐观。适合描述 AI 工具、外包团队等。
8. X is one of the few places left where you don't get Y
“It's one of the few places left where you don't get instant answers.”
「所剩不多的……之一」句式，暗含对时代变迁的感慨。where 引导定语从句说明该场所的特征。适合描述某种正在消失的价值。
9. often what you're doing is that you're removing X until there's only Y left
“Often what you're doing is that you're removing the negative space of all the wastes of the problem that don't work until there's only one path left.”
重新诠释「看似无进展」的过程：what you're doing is… 把表面行为改写为真实意义，until 交代终点。适合解释隐性进步。
生词精讲 · 118 · 按出现顺序
food for thought
phr.
0:00
值得思考的东西、引人深思的材料
interchange
/ˈɪntərˌtʃeɪndʒ/
n.
0:00
交流、互换（思想）
promoted to full professor
phr.
0:34
晋升为正教授（美国教授职级最高一级）
have a handle of
phr.
1:59
对……有把握/了解（常作 have a handle on）
precious
/ˈpreʃəs/
adj.
1:59
珍贵的、宝贵的
median
/ˈmiːdiən/
adj./n.
2:39
中位数的；此处指「最典型的、居中的」反应
cast
/kæst/
v.
2:39
施（法术）；cast a spell on sb 对某人施咒
juggling
/ˈdʒʌɡlɪŋ/
v.
2:39
同时应付（多件事）；本义为杂耍抛接
archetypes
/ˈɑːrkɪˌtaɪps/
n.
3:26
原型、典型形象（此处指刻板印象）
in one go
phr.
3:26
一次性、一口气
outreach
/ˈaʊtriːtʃ/
n.
4:31
外展、科普推广活动
ivory tower
/ˈaɪvəri ˈtaʊər/
n.
4:31
象牙塔，脱离现实的学术环境
internalize
/ɪnˈtɜːrnəˌlaɪz/
v.
4:31
内化（使成为自身的一部分）
pre-rigorous
/ˌpriːˈrɪɡərəs/
adj.
4:31
前严格的（陶哲轩自创术语，指靠直觉而非证明的阶段）
dreaded
/ˈdredɪd/
adj.
5:32
令人畏惧的、闻风丧胆的
pooh-poohed
/ˌpuːˈpuːd/
v.
5:32
对……嗤之以鼻、不屑一顾（口语）
fluidly
/ˈfluːɪdli/
adv.
6:31
流畅地、灵活地
wave your hands
phr.
6:31
（数学口语）挥手带过，做不严格的粗略论证（hand-waving）
proof by contradiction
n. phr.
6:31
反证法
recess
/ˈriːses/
n.
6:31
（学校的）课间休息
quadrillion
/kwɑːˈdrɪljən/
n.
6:31
千万亿（10^15）
verbalize
/ˈvɜːrbəˌlaɪz/
v.
7:34
用言语表达出来
come in various flavors
phr.
8:14
有各种不同的类型/形式
hypotheses
/haɪˈpɑːθəˌsiːz/
n.
8:14
假设（hypothesis 的复数）
cancel terms
phr.
8:48
约掉（等式两边的）项
irrational number
n. phr.
8:48
无理数
ratio
/ˈreɪʃioʊ/
n.
8:48
比、比值
a move in a game
phr.
10:01
棋局中的一步；引申为「常规手段之一」
reductio ad absurdum
/rɪˌdʌktioʊ æd æbˈsɜːrdəm/
n.（拉丁）
10:33
归谬法、反证法
gambit
/ˈɡæmbɪt/
n.
10:33
（象棋）弃着、开局让子；引申为策略性开局
bishop
/ˈbɪʃəp/
n.
10:33
（国际象棋）象
rook
/rʊk/
n.
10:33
（国际象棋）车
positional advantage
n. phr.
10:33
（棋类）局面优势
sign error
n. phr.
11:06
正负号错误
disciplines
/ˈdɪsəplɪnz/
n.
11:06
学科、领域
mindset
/ˈmaɪndset/
n.
11:53
思维模式、心态
disconnect
/ˈdɪskəˌnekt/
n.
12:41
脱节、不一致
Taylor approximation
n. phr.
12:41
泰勒逼近（用多项式近似函数）
paralyzed
/ˈpærəˌlaɪzd/
adj.
13:19
瘫痪的、动弹不得的（此处指无法行动）
analysis paralysis
/əˈnæləsɪs pəˈræləsɪs/
n. phr.
13:19
分析瘫痪：因过度分析、选项过多而无法决策
unblocked
/ˌʌnˈblɑːkt/
v.
13:19
解除阻塞、让……重新前进
narrow field
n. phr.
14:13
狭窄的领域
wrong turns
n. phr.
14:42
走错的弯路、错误的尝试
default state
n. phr.
14:42
默认状态、常态
impostor
/ɪmˈpɑːstər/
n.
14:42
冒名顶替者；impostor syndrome 冒名顶替综合征
normalize
/ˈnɔːrməˌlaɪz/
v.
15:34
使正常化、使被普遍接受
partial differential equations
n. phr.
15:34
偏微分方程（PDE）
cocky
/ˈkɑːki/
adj.
15:34
自负的、过分自信的
write up
phr. v.
15:34
整理成文、撰写（正式稿）
get rid of
phr. v.
16:23
消除、摆脱
invested
/ɪnˈvestɪd/
adj.
16:50
（情感/精力上）投入很深的
kept at it
phr.
16:50
坚持不懈地做下去（keep at sth）
corollary
/ˈkɔːrəˌleri/
n.
17:20
推论、必然结果
condition
/kənˈdɪʃən/
v.
17:20
训练、使习惯于（某种行为）
speak up
phr. v.
17:20
大胆发言、开口说话
static
/ˈstætɪk/
adj.
18:03
静态的、一成不变的
make it your own
phr.
18:03
把它变成自己的（真正吸收、内化）
inequality
/ˌɪnɪˈkwɑːləti/
n.
19:04
（数学）不等式
end point case
n. phr.
19:04
端点情形（参数取边界值时的情况）
disprove
/dɪsˈpruːv/
v.
19:40
证伪、反驳
counterexample
/ˈkaʊntərɪɡˌzæmpəl/
n.
19:40
反例
occurred to me
phr.
19:40
（想法）浮现在我脑海中（It never occurred to me that…）
embrace
/ɪmˈbreɪs/
v.
20:06
欣然接受、拥抱（观念）
echoed
/ˈekoʊd/
v.
20:50
被呼应、被重复
perseverance
/ˌpɜːrsəˈvɪrəns/
n.
20:50
坚持不懈、毅力
habits of mind
n. phr.
20:50
思维习惯（教育学术语）
solitary
/ˈsɑːləˌteri/
adj.
22:08
独自的、孤独的
interdisciplinary
/ˌɪntərˈdɪsəplɪˌneri/
adj.
22:08
跨学科的
strong suit
n. phr.
22:43
强项、擅长之处
division of labor
n. phr.
22:43
分工
for better or for worse
phr.
23:46
不管是好是坏、无论结果如何
endowment fund
/ɪnˈdaʊmənt fʌnd/
n.
24:24
（大学）捐赠基金
crush
/krʌʃ/
v.
24:24
（口语）碾压、远远超过
asterisk
/ˈæstərɪsk/
n.
25:08
星号；引申为「附带说明/例外条件」
risk-benefit analysis
n. phr.
25:08
风险收益分析
portfolio
/pɔːrtˈfoʊlioʊ/
n.
25:49
投资组合
push the boundaries
phr.
25:49
突破边界、推进前沿
dig up
phr. v.
25:49
挖掘出、找出（资料）
fall flat on their face
idiom
25:49
彻底失败、栽大跟头
success rate
n. phr.
26:41
成功率
MO
/ˌem ˈoʊ/
n.
27:20
modus operandi 的缩写，惯常做法、行事方式
fluid
/ˈfluːɪd/
adj.
27:20
不稳定的、变动不居的（局势）
prodigy
/ˈprɑːdədʒi/
n.
27:59
神童、天才儿童
Fast forward to
phr.
27:59
快进到（某时间点）
terrified
/ˈterəˌfaɪd/
adj.
27:59
极度害怕的
accelerated
/əkˈseləˌreɪtɪd/
adj.
29:34
加速的、超前的（教育）
headmasters
/ˈhedˌmæstərz/
n.
29:34
（英式/澳式）中小学校长
improvising
/ˈɪmprəˌvaɪzɪŋ/
v.
29:34
即兴发挥、临时应对
rigid
/ˈrɪdʒɪd/
adj.
30:13
死板的、僵化的
get away with
phr. v.
30:13
侥幸做成（本不被允许的事）、蒙混过关
showcase
/ˈʃoʊkeɪs/
v.
30:58
展示、陈列
first generation students
n. phr.
31:36
第一代大学生（家中首位上大学的人）
speaking with one voice
phr.
32:34
用同一个声音说话、口径一致
post secondary
/ˌpoʊst ˈsekənˌderi/
adj.
32:34
高中后的、高等教育阶段的
compactness
/kəmˈpæktnəs/
n.
33:25
（拓扑学）紧致性
topology
/təˈpɑːlədʒi/
n.
33:25
拓扑学
checkable
/ˈtʃekəbəl/
adj.
34:10
可检验的
verifiable
/ˈverəˌfaɪəbəl/
adj.
34:10
可验证的
hit gold
idiom
34:51
撞上金子、意外获得重大成功
rubbish
/ˈrʌbɪʃ/
n.
34:51
（英式）垃圾、废话
rush it
phr.
35:36
操之过急、匆忙行事
outtakes
/ˈaʊtteɪks/
n.
36:09
（影视）NG 片段、被剪掉的镜头
taking the lid off
idiom
36:55
揭开盖子、揭示内幕
under the hood
idiom
36:55
引擎盖下面；引申为内部运作机制
pedagogic
/ˌpedəˈɡɑːdʒɪk/
adj.
38:06
教学法的、教育学的
walk-through
/ˈwɔːkθruː/
n.
38:57
（游戏）攻略、逐步指南
brainwave
/ˈbreɪnweɪv/
n.
39:36
（英式口语）灵光一闪、突然的好主意
earned
/ɜːrnd/
adj.
39:36
应得的、靠努力挣来的
negative space
n. phr.
39:36
负空间（艺术术语，主体之外的空白区域）；此处引申为排除掉的无效路径
extraneous
/ɪkˈstreɪniəs/
adj.
40:26
无关的、外来的
by the same token
phr.
40:26
同样地、出于同一理由
instantaneous
/ˌɪnstənˈteɪniəs/
adj.
40:26
瞬间的、即时的
suck up
phr. v.
40:57
吸收、吞进（数据）
training data
n. phr.
40:57
训练数据
remix
/ˈriːmɪks/
n.
41:55
重新混合、混搭改编
stunned
/stʌnd/
v.
41:55
使震惊、使目瞪口呆
under debate
phr.
42:34
尚有争议、仍在讨论中
fielding
/ˈfiːldɪŋ/
v.
42:34
应对、回答（一连串问题）
理解自测 · 11 题 · 是真懂了，还是以为自己懂
1. 陶哲轩把数学思维分为哪三个阶段？各大致对应什么教育阶段？
三个阶段是前严格（pre-rigorous）、严格（rigorous）和后严格（post-rigorous）。前严格阶段大致对应 K-12 教育，学生通过例子、直觉、公式和计算学习，允许犯错，对概念只有模糊把握；严格阶段对应本科数学专业的证明课，只有一种正确解法，旧有直觉被「嗤之以鼻」；后严格阶段对应研究生阶段，此时可以流畅地非正式推理，同时知道随时能把它转化为严格证明，在两者间自由切换。陶强调后严格阶段才是「有趣的部分」，但多数人从未到达。
2. 陶哲轩用什么童年游戏说明小学生已经自己「发明」了反证法？
「谁能说出最大的数」游戏。孩子们轮流报出十亿、万亿、千万亿等越来越大的数，直到有人意识到：无论对方说什么数，下一个人总能说「那个数加一」。由此他们领悟到不存在最大的数——因为如果存在，它就会比自己大，这不可能。陶指出这正是反证法的完整逻辑，孩子们已经掌握了概念，只是没有语言把它「说出来」，这就是前严格阶段的典型状态。演讲第二部分用这个例子对照本科课堂上「√2 是无理数」的正式反证，说明同一概念在不同阶段的不同面貌。
3. 关于 Bourgain 相关的那个偏微分方程问题，陶的团队犯了什么错误？最终结果如何？
团队把某个表达式展开成 13 项，正确控制了其中 12 项，却漏掉了第 13 项——而这恰恰是最难处理的一项。他们当时以为几个月就能完成，甚至已在预订餐厅准备开香槟庆祝，直到一位更细心的合作者发现遗漏。检查后发现无论如何都无法消掉这一项，六个月投入付诸东流，庆祝取消。但因为投入已深，团队坚持了两年，最终用完全不同的方法解决了问题，论文发表在《数学年刊》并获奖，成为陶最引以为傲的成果之一。
4. 陶对当前大语言模型做高等数学的能力给出了怎样的评估和数据？
陶把当前 LLM 比作「一支水平参差不齐的研究生大军」：给它们一个问题，它们会从文献中翻出各种技巧逐一尝试，有时表现出色，有时栽大跟头。他特别提醒，社交媒体上只展示最成功的案例，看起来惊人，但实际把这些工具用在自己关心的问题上，成功率往往只有 1% 到 2%。在问答后段他补充：AI 已解决过一些此前未解的问题，但方法多是「相关问题方法的混搭」，尚未出现真正让他震撼的全新证明；在生成定义这类无法打分验证的任务上，AI 仍「很让人抓狂」。
5. 为什么陶认为把数学家塑造成「天才」的叙事会伤害学生？这一论点与演讲后面的哪个主题相连？
陶的推理链是：如果学生被灌输「数学家都是快乐的天才」，那么当他们自己解题一次做不出来、或某个难点怎么也学不会时，就会推断「我不是天才，所以我不是当数学家的料」，进而在数学变得有趣之前放弃。这一论点直接连接到演讲第二大主题「失败的自由」：天才叙事隐含「真正的数学家不会失败」，而陶用自己六个月错误、Bohr 的专家定义和 Mirzakhani 论文「只有胜利」的例子说明，失败与迷失才是数学家的默认状态。修正学生自我认同的关键，就是让他们看到数学家真实的失败过程。
6. 陶说「数学中失败很廉价」，但在回答捐赠基金问题时又加了「附注」。这个限定说明了什么？
陶的原论点是：与工程师设计桥梁、外科医生手术相比，数学证明失败的代价只是「重来一遍」，因此研究生被教导要不断尝试、不断犯错。但当提问者建议数学系用 AI 管理 UCLA 五十亿美元捐赠基金时，陶立刻指出「失败廉价」有一个星号：一旦数学与现实世界连接（如金融），就必须做严肃的风险收益分析。他建议用模拟市场做教学项目，而不是真的让学生管钱。这说明「失败的自由」是纯数学内部的结构性特征，源于错误不产生外部后果，并非可以无条件推广到应用场景的普遍原则。
7. 从 Stein 不等式的经历中，陶得出了关于「蠢问题」的什么结论？为什么说这是对「失败的自由」的推论？
陶多年试图证明导师 Stein 不等式的端点情形，只得到很弱的部分结果。在一次会议上，一位同事问：「你试过证伪吗？有理由相信它成立吗？有反例吗？」陶从未想过这个命题可能为假，一周内就找到了反例。他的结论是：不一定非得你自己问蠢问题，但「总得有人问」——把问蠢问题从个人习惯提升为群体机制。这是「失败的自由」的推论（corollary），因为敢问蠢问题的前提是不怕显得愚蠢，正如敢尝试的前提是不怕失败；教学中「不确定就别开口」的训练与「错一个符号就扣分」同样压制了探索。
8. 陶为什么用「负空间」来解释做数学的感觉？这个比喻回应了什么常见误解？
陶用 90 年代没有网络攻略的电子游戏作比：每晚卡在同一扇门前，直到某天灵光一闪，搬开一块石头找到钥匙。他解释说，看似毫无进展的那些日子，实际上是在清除「负空间」——问题中所有走不通的死路——直到只剩一条路，此时突破自然到来。这回应了「灵光一闪」是天赋或运气的误解：事后觉得「我怎么之前没看出来」，是因为之前还没下够功夫清理无关的杂乱。这也呼应了 Bohr 的「专家是犯过所有错误的人」，并解释了为何解答「靠自己挣来」的满足感来自前期的失败积累。
9. 陶认为 AI 在哪类数学任务上有用、在哪类上无用？这一区分的依据是什么？
陶的区分标准是「可验证性」。对于可检验的任务——问题解出与否一目了然——AI 目前相当有用，尤其当一个大任务能拆成上百个子问题、且可以接受工具只解决其中 10% 的时候。但对于像「创造一个好定义」这类无法打分的任务，AI 表现令人沮丧：它可能生成一个好定义和九个坏定义，自己无法分辨，偶尔「撞上金子」也一带而过。他因此主张数学要「工业化」、「玩百分比游戏」，改变传统一人死磕一题的模式。这一区分的深层依据是：AI 缺乏对自身产出价值的判断力，而这种判断力恰恰是数学创造力的核心。
10. Kate Stevenson 指出陶的三点框架可能缺失了什么？陶会如何回应这一补充？
Stevenson 指出，面对没上过数学夏令营、高中不算成功的第一代大学生和背景艰难的学生，「能够犯错并感到安全」本身是一种特权，教师必须先建立信任，学生才愿意去犯错。她主张教育界「用同一个声音」告诉学生数学就是关于犯错与坚持，是思想之网而非算法堆砌，并提到 CSU、UC 与社区学院已联合发表相关声明。从陶的立场推断，他会认同这一补充：他本人承认自己受教育的年代「更慢、更可预测」且大家为他临场发挥，也承认「不知道什么管用」；他的「失败廉价」讲的是数学的内在结构，而信任问题属于教学环境的外部条件，两者互补而非冲突。
11. 如果把「失败廉价、允许多试」的数学思维迁移到软件工程或创业等领域，还成立吗？陶的框架给出了什么判断标准？
部分成立。陶的核心判断标准是失败是否产生外部代价：数学证明失败只需重来，桥梁与手术失败代价高昂，金融操作则「必须做风险收益分析」。按这一标准，软件工程中有测试环境、可回滚的代码变更接近数学（失败廉价，应鼓励快速尝试），而生产环境事故、涉及用户数据的操作则接近工程师造桥。创业中「小成本快速试错」符合陶的逻辑，但押上全部资金则否。此外，陶在 Arnold 引文后自嘲「考虑到现在的 AI……算了」，暗示即使在数学内部，AI 算力成本也在改变「实验廉价」的前提。因此迁移时应先问：这次失败的代价由谁承担、能否轻易复原。
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