丹尼尔·利特：数学家的AI指南

然后突然之间，你就有了这些机会，一个新想法被提出来，然后你就能一下子解决一堆之前不明白的问题。

And then suddenly, you know, you have these opportunistic situations where a new idea has been introduced, and you can kind of now find you now can suddenly, like, cascade through a bunch of other things that we didn't understand before.

对。

Yeah.

我的意思是，那回到那些三、四、五年的问题，模型在哪些地方不太有用呢？

I mean, so then going back to the three, four, five year problems, and where are the models sort of not useful?

换个方式问吧。

Kind of ask it another way.

比如，当你深入思考的时候，是不是只是问题还没被明确地表述出来，而更多的是你在思考那些根本的物理原理，你知道的，就是那些……

Like, when you're doing the deep thinking, like, is it just that it's just not clear that you formulated as a problem, and it's more that you're thinking about these like, what are, you know, what what are the fundamental kind of physics of, you know, of

对。

Yeah.

所以在某些情况下，我的意思是，有些情况下，问题确实被很好地表述出来了。

So in some cases, I mean, the the some cases, there is, like, a well stated problem here.

比如，你知道，我说过我已经思考某些问题大概十年了，到现在还是这样。

Like, know, I I said I've been thinking about things for, you know, maybe ten years now at this point for certain problems.

有时候，就是有一个我想证明的猜想，它就在那里。

Sometimes there is just like a conjecture that I would like to prove that that is is there.

我觉得模型在某些问题上可能帮不上忙的一个原因是，这些猜想是真的。

I think one reason the models might not be useful for some of these things is, like, the conjectures are true.

明白。

Mhmm.

所以，比如说，那个单位距离问题，大家普遍认为它是真的，结果后来发现是假的。

So, like, I think that certain you know, for for example, you know, this unit distance problem, like, the the general belief in the community was that it was true, and then it turned out to be false.

明白。

Mhmm.

所以这意味着，你可以做一个特定的构造来……

And so what that means is that there's, like, a specific construction you can do to

明白。

Mhmm.

来反驳它。

To to to refute it.

另一方面，我觉得我思考的很多事情，怎么说呢，我也不太确定。

On the other hand, I think a lot of the things I I think about, like I don't know.

好吧。

Okay.

也许我马上就要，你懂的，明天就有人会提出一个反例来推翻那个p曲率猜想假设，那我可就丢人了。

May maybe I'm about to you know, this someone will come up with a founder example to the p curvature conjecture hypothesis tomorrow, and, like, I'll look like a fool.

但总的来说，这些猜想都符合一些非常广泛的理论框架，这意味着，你看，我们其实有很多证据表明它们是对的。

But they in general, like, these conjectures fit into some very broad theoretical framework, which means that, look, we actually have a lot of evidence that they're true.

所以，是的，这也是其中的一部分原因。

And so, you know, yeah, so that's part of it.

就是说，并没有一个现成的构造可以直接推翻它。

Like, there's not, like, a construction you can do to refute it.

你需要某种方式，你懂的，我们有这个庞大的框架，其中某些部分只是猜想，你可能需要解决其中一些猜想才能赢。

You need to somehow kind of you know, there's we have this giant framework where certain certain pieces of it are only conjectured, and you probably need to to resolve some of those conjectures to win.

而且，我觉得我们也比较清楚，要解决这些猜想，确实需要非常全新的思路。

We also have, like, a pretty good sense, I think, that, like, very serious new ideas are needed to resolve those conjectures.

所以，当然，你没法确定。

So, like, of course, you can't be sure.

比如，也许有一些非常巧妙的构造方法，能让你，怎么说呢，避免需要一个大新想法。

Like, maybe there's some very clever construction that will let you, I don't know, avoid having a big new idea.

对。

Mhmm.

我不知道。

I don't know.

也许，你知道，我们很有可能发现这一点。

Maybe, you know, it's quite possible we'll find this out.

但我的感觉是，至少对于我一直在思考的很多问题来说，它们就是无法通过，你知道，以非常技术性的方式应用现有技术来解决。

But my sense is that for at least a lot of the things I've been thinking about, there are they're just not accessible to, you know, applying known techniques very, very tech in a very technically strong way.

所以你需要开发一种新技术。

So you need to develop a new technique.

我只是想说明白，我，好吧。

I'm just to be clear, I okay.

不是说模型做不到这一点。

Not saying the models won't be able to do this.

只是到目前为止，它们似乎还做不到。

Just, like, so far, they seem not to

其实，不是的。

Actually, no.

这正是我想深入探讨的点，因为，你知道，正如你所说，就是，好吧。

That that's exactly the point that I wanted to delve into because it's you know, she point to your point, it's like, okay.

我们可以尝试校准和预测，比如为什么它们会在这方面变得更好，但事实确实如此。

We can just try try to calibrate and forecast, like, why they would get better at this, but it is true.

比如，你知道，就是提供反例的构造，它们似乎很擅长。

Like, you know, that just providing construction for a counterexample, they seem to be strong on.

但如果要开始发展新的理论，或者像你说的，新的技术、工具来解决，来证明猜想为什么是对的，它们就更加吃力了。

If you have to start developing either new theory or to your point, techniques, machinery to solve, to to prove why conjecture is true, it struggles more.

这可能是因为它们所依赖的很多东西，其实也是其他领域已有的技术，然后它们把这些技术移植过来。

And it's probably because a lot of what it's drawing on is also just, like, techniques that have happened in other areas, and they're porting it over.

而且就像你说的，这可能就是为什么单位距离问题是一个更有创意的结果，因为它可能更多是在自己独立完成这些。

And and to your point, that's why maybe the unit distance problem was such a more creative result because it was maybe doing more of that on its own.

像是从另一个领域来的。

Was, like, from another.

至少它像是从某个意想不到的领域借鉴了东西。

At least it was, like, drawing in something from an unexpected area.

对。

Yeah.

没错。

Exactly.

没错。

Exactly.

所以，也许接下来要问的问题是，并不是因为现在我们觉得，好吧。

And so, like, maybe then to kinda ask the question, it's not that because now we're like, okay.

行。

Fine.

人工智能进步得太快太猛了。

AI is getting so good so fast.

我们不能小看它。

We can't count it out.

但是，你觉得它需要什么呢？

But, like, why what do you think it has to yeah.

我想你是在说它需要改进的地方，但也许可以再多谈谈，为什么发展出那种新理论和新技术会特别困难。

I guess it's it's you're spelling out what it has to get better at, but maybe some more kind of feelings on, like, why it's kind of particularly hard to then develop that new theory and technique.

这是个好问题。

It's a good question.

我觉得你需要不同的——其实我猜这完全有可能做到。

And I think you just need a different like, my my guess actually is it's probably totally doable.

只是还没人做出来而已。

It just hasn't been done yet.

也许你只需要一个不同的强化学习环境。

Like, maybe you just need a different RL environment.

我不知道。

I don't know.

对。

Yeah.

所以我的意思是，到了这个阶段，我的预期就是这种轨迹会继续往上走。

So I mean, at this point, my expectation is just, like, that the trajectory will continue upwards.

我并不怀疑能力会持续增长。

I'm not skeptic of continued capabilities growth.

对。

Yeah.

但是，是的，你知道，我觉得肯定的一点是，像发展理论或者理解某个不太清楚的对象这种技能，是比较模糊的。

But, yeah, you know, I think what is definitely true is that, like, skill of, like, developing a theory or, you know, like, building your understanding of some poorly understood object is, like, a fuzzier one.

明白。

Mhmm.

所以可能更难。

So it might be harder.

你知道吗？

You know?

我觉得你可以试着让它去发展你对泽塔函数的理解。

I guess you can try you can tell it, you know, develop your understanding of zeta functions.

然后一旦它通过了你的假设，你就给它奖励。

Then once it proves through your hypothesis, you give it a reward.

但我觉得更难的是想出一些中间的、可以奖励的东西。

But it's, like, kinda harder, I think, to come up with some, like, intermediate No.

可以奖励的东西。

Things that you can reward.

对。

Yeah.

对。

Yeah.

不过话说回来，我确实认为数学整体上提供了很多难度不同的猜想。

I mean, that said, you know, I do think mathematics as a whole provides a lot of conjectures of varying levels of difficulty.

对。

Mhmm.

所以，这也许能解释为什么这些领域看起来有点进展。

So, you know, maybe maybe this explains why there seems to be a little bit of progress in these areas.

比如，他们大概是想让它解决很多问题，通过解决这些问题至少培养一些理论方面的技能。

Like, presumably, they are trying to, you know, get it to solve lots of problems and solve those problems develop at least some of the skills of theory.

人类能够培养这些技能。

Humans are able to develop these skills.

你知道，有时候他们会从博士导师那里得到奖励，导师会说，哦，这是个好主意，或者基于某种人类品味之类的东西。

You know, I guess sometimes they get rewards from their PhD advisers, and their adviser says, oh, that's a good idea or something based on some element of human taste or whatever.

是的。

Yeah.

这也许也是可以做到的。

And that that might be something one can do too.

但是，是的，我的预期是，随着能力的持续增长，我们也会看到这些领域的进步。

But, yeah, my my my expectation is that just, like, as part of continued capabilities growth, we'll see growth in these areas too.

对。

Yeah.

对。

Yeah.

而且我喜欢你这种框架，就是把越来越难的猜想当作一种课程，既适合人类，也适合AI。

And I like the framing where you're sort of casting these increasingly difficult conjectures as a as a, you know, a form of curricula for both humans and, of course, AI.

这可能会让一些人觉得有点太哲学了。

And it it you know, and maybe kind of getting a little bit too philosophical for some people's taste.

这其实也触及了一个问题，就是为什么我们在数学上特别擅长，或者也可能特别不擅长？

It's like it it it gets at the question of, like, why are we particularly good or maybe particularly bad at math too?

因为，你知道，当我们发展新理论的时候，我们到底是在挣扎做什么，或者有些人特别擅长什么？

Because it's you know, what what is it that we're either struggling to do or or some people are particularly good at when you develop new theory.

因为这又不完全是，或者说可能跟为什么我们能构建好的物理学结构有关。

Because it's it's not again, it's, like, not really or maybe it's related to, like, why can we formulate good structures for physics as well.

就是，这并不明显。

It's just like it's not obvious.

世界的各个部分在某种程度上都是可以理解的，也是可以这样解读的，但有些部分不是。

All parts of the world are kind of understandable, and legible that way, but some parts are.

因此，我们试图去理解，因为我们的思维可能承受着压缩的压力，因为我们无法理解任何超出更精细压缩范围的东西。

And therefore, we try to do it because there's maybe compressive pressures on our our, you know, our our minds because we need to, we can't understand anything, beyond, you know, compressing, stuff, more finely.

我不知道这是否也是正确的解释，

I don't know if that's also, like, the correct interpretation of

我觉得那是

I think that's

对。

Yeah.

我有点怀疑把压缩作为兴趣的衡量标准，但作为人类学上的原因，解释我们为什么构建某种理论，这绝对是成立的。

That's I'm not I I'm a little bit skeptical as, like, compression as a metric of interest, but it's definitely, like, as a anthropological reason for, like, why we do a certain theory in building.

我只是觉得这是事实。

I just think it's true.

事实上，我认为我们无法只是埋头苦干，这一点对我们做出发现的能力很重要。

And in fact, I think it's, like, kind of like, our our inability to just grind is kind of important to our ability to make discoveries.

举个例子，我目前有一篇论文，里面的模型还挺有用的。

So and just, like, as an example so I've I have one paper out so far where the models were kind of useful.

它们帮我证明了一些引理。

So they, like, like, proved a couple lemmas

在论文里。

on paper.

当时的情况是，我们已经证明了主要结果，但有些引理我不太满意。

So this is some some situation where, like, we had kind of proven the main result, and then there were, like, some lemmas I was unhappy with.

它们看起来不是最优的，所以我就和Gemini DeepThink Oh合作改进。

Like, they seemed not to be optimal, and so I, like, worked with actually, this was with with Gemini DeepThink Oh.

那时候它还在前沿，现在已经不是了。

Which at the time was also on the frontier, no no longer.

我主要靠它来改进这些引理。

I I where I kinda worked with it to to improve the lemmas.

结果呢，我想证明某个引理，但模型做不到。

And so what happened here was there was some lemma I wanted to prove, and the models couldn't do it.

所以，所有前沿模型都做不到。

So the none none of the frontier models could do it.

然后我自己算了一大堆例子，突然意识到，哦，也许这里有个原因说明它可能是对的。

And so I, like, worked out a ton of examples on my own, and I realized, oh, well, maybe, like, here's some reason why it could be true.

然后我找到了一个更好的表述。

Like, so I, like, I found a better statement of.

然后，一旦有了那个表述，我自己可能很快就能证明，但模型也能很快证明那个更好的表述。

And then, okay, once I had that statement, like, okay, probably I could have done it pretty fast, but also the models were able to very quickly prove this prove prove that sort of better statement.

所以，我们无法证明它，反而让结果得到了改进。

So, like, our inability to prove it, like, led to an improvement in the result.

好的。

Okay.

现在，你可以把原来那个模型证明不了的引理放进ChatGPT 5.6 Pro，它会输出你见过的最糟糕的证明，大概十页，全是硬算，毫无洞见。

Now you can take the original lemma I had that the models weren't able to prove, put it into ChatGPT 5.6 Pro, and it will output, like, the worst proof you've ever seen, like, 10 pages of, like, just brutal calculations with no insight whatsoever.

所以，这个证明本身没问题，但它不会引导我们发现那个我认为很美的概念性解释，来解释我们发现的这个事实为什么成立。

And so, okay, this would have been a perfectly fine proof, but it would not have led to this discovery of, like, I think, kind of beautiful conceptual explanation for why this thing we discovered was true.

所以，我们因为做不出来，反而找到了一个更好的证明，对吧。

So, like, we found a better proof because we couldn't do mean, okay.

我说我们做不出来，这事儿确实发生了。

And I I say we couldn't do the, like, actually happened.

而且我知道，我之前抱怨那些难看的证明，现在自己也有点双标了。

And here, I I understand that I'm being hypocritical about my complaints about UglyProofs before.

对，就是这样。

It's like Yeah.

我意识到，那种枯燥的硬算证明其实能行。

I realized, like, this horrible grind proof would work.

对。

Yeah.

但我就是没法逼自己去做。

And I, like, could not bring myself to do it.

所以我就换了个思路。

And so I, like, put to another argument.

所以，我的意思是，也不是那样。

And so, I mean, it's not yeah.

关键在于把它做出来。

It's about doing it out.

这些模型能相当可靠地完成很长的技术计算。

The models can do a very long technical calculations pretty reliably.

对。

Yeah.

嗯，我是说，这得归功于你。

Well, I mean, to your credit

我不知道。

I don't know.

我想我理解你为什么——就是，别因为某个证明看起来难看就回避去证明它。

I think I understand why you're it's like, don't, you know, just shy away from trying to prove something just because it seems ugly.

因为就像你得迈出第一步。

Because it's like you have to take the first step.

对吧？

Right?

而且最终，你都会努力去追求更深的理解。

And, eventually, you will all try to work towards insight.

所以，你可能不好意思承认，但驱动你的可能还是审美，或者纯粹就是——我想弄明白。

And so, you know, you might be shy about admitting it, but it is still maybe driven by whether it's, like, aesthetics or just, like, pure you know, it's like, I wanna understand.

如果理解就意味着稍微简单一点，或者更紧凑一些，那我已经理解了。

And if understanding just means it's a little bit simpler or, you know, more compressed than than I've gotten understanding.

这其实也是一个很深的哲学问题。

I mean, that's just like a deep philosophical question too.

就像，理解到底是什么？

It's like, what it is?

对。

Right.

当然，我们做非正式数学而不是写一长串正式的CFC符号串，肯定是有原因的。

Like, mean, of course, any you know, there's a reason we do informal mathematics rather than, like, writing out long and formal strings of symbols of CFC or whatever.

对。

Yeah.

对。

Yeah.

尽管除了长度之外，我们其实是在用一种方式，试图从中获得一些非严格的、直觉上的理解。

Even though, you know, besides length, it's just somehow we're trying to put it in a way where we are actually getting some nonrigorous understanding out.

对。

Yeah.

我的意思是，而且——

I mean, and

我觉得某种程度上。

I think somehow.

对。

Yeah.

而且这甚至关系到，为什么这样会有帮助。

And even how that even relates to, like, why it helps.

而且，就像，是不是因为我们不能死磕，是不是因为，如果我们朝着更压缩的方向努力，那它也有望在其他领域上提升，有助于理解其他领域？

And, it's like, is is it because of our inability to grind, is it because, like, if we if we are pushing towards something that's more compressive, it also hopefully also ascends on the does it help understand other fields as well?

而且，当我们试图同时优化这两个目标时，它们往往重合，这确实有点神奇。

And it's it's just it is somewhat magical that when we try to optimize for two the the both, that it it tends to coincide.

我不知道这是否公平，这其实是一个定量的说法。

I don't know actually know if that's a fair it's it's actually a quantitative statement.

但是，为什么，你知道，这有点神奇。

But, like, why, you know, that is is is kind of magical.

至少有时候是这样。

It's at least sometimes true.

对。

Yeah.

对。

Yeah.

没错。

Exactly.

或者说，你知道，我们确实偏向于那些成立的情况。

Or or, you know, we we certainly bias towards the cases of which it is.

这就是为什么我们说这是一个值得构建的好理论。

That's why we call that a good theory to build.

但确实，是的。

But it's it's, yeah.

我觉得这很大程度上，你知道，就像这是我们可以研究和理解的，因此，那里有丰富的数学成果，也正好很方便。

It's very much, I think, you know, it's like this is what we can study and understand, and therefore, it's also very convenient that it was rich, in in kind of mathematical results there.

对。

Yeah.

我是说，好吧。

I mean okay.

所以我觉得有两件事我们可以，你知道，去探讨，那就是，既然你相信AI的数学能力会持续进步，并且能做你现在做的这些事情，那么数学界应该如何最好地适应并从中受益呢？

So I think there's two things that we can, like, you know, pull on, which is like what, given you've seeded that or not seeded, but you've believed that mathematical ability of AI is gonna continue advancing and can do some of the things that you're, you you know, doing nowadays.

数学界应该如何最好地适应并从中受益呢？

How how should, like, the mathematics community kind of best, adapt and benefit from this?

作为一个已经没有时间再练习数学的人，我觉得这真的很棒，因为我可以多涉猎一些。

I mean, as a just kind of saying as somebody who's you know, doesn't have the time to kind of practice mathematics anymore, this is kind of very great because I can maybe dabble more.

会有很多成果涌现出来。

There's a lot of results that can come out.

但我也能看到，你更精确地指出了我们无法激发正确的理解和发展的行为。

But I can also see where, you know, you've made the more precise point of we can not motivate the right kind of behavior of understanding and development.

所以我很想听听你在这方面的看法。

And so I would love to kind of hear more about your views there.

对。

Yeah.

好。

So okay.

首先，这确实非常令人兴奋，因为有越来越强大的模型在产生高质量的结果。

So first of all, like, it is clearly really exciting, like, that there are increasingly capable models that are, like, producing high quality results.

至少，你知道，一些高质量的结果。

At least, you know, some high quality results.

而且也有很多思考。

Also also a lot of thought.

对。

Yeah.

对。

Yeah.

有些不错的东西。

Some some good stuff.

所以，你知道，随着模型变得越来越强大，我希望它们能回答很多我一直在思考、甚至让我夜不能寐的问题。

And so, you know, as the models get really capable, you know, my hope is that they will answer a lot of the questions that I've been, like, you know, kept up at night thinking about.

我想，我就能知道答案了。

I'm like, I'll I'll get to learn the answers.

我觉得这真的很令人兴奋，你知道，很多人进入数学领域，很大程度上是因为他们喜欢学习数学。

Think that's really exciting, you know, a lot of people got into math, like, largely because they enjoyed learning math.

对。

Yeah.

但是，作为数学学生，你首先做的事情就是学习别人已经做过的东西，而且我觉得你得花大概二十年时间才能开始做自己的研究，哦，也许不用整整二十年。

But, like, the first thing you do as a a math student is you, like, learn stuff that other people did, and you do that for, like, twenty years, I think, before you before you start doing oh, maybe not quite twenty years.

但你知道，十五年吧。

But, you know, fifteen years.

如果幸运的话，二十年。

If you're lucky, twenty years.

对。

Yeah.

你开始得早。

You started early.

对。

Yeah.

对。

Yeah.

对。

Yeah.

所以话说回来，数学的目标并不是为了发表数学论文。

So that said, like, the goal of mathematics is not to produce mathematics papers.

而是为了产生某种理解。

Like, it's to produce some kind of understanding.

也许有些理解会存在于模型的权重里之类的。

So maybe, okay, maybe that some of that understanding resides in model weights or something.

但对我来说，那挺让人不满意的。

To me, that's, like, pretty unsatisfying.

对。

Like Mhmm.

我自己做数学的动机，就是满足我个人的好奇心。

My own, you know, my motivation for doing mathematics is, like, I would like to satisfy my own personal curiosity.

我觉得人们应该有这个能力去做这件事。

I think people should be have the capability to do that.

而这需要一套相当庞大的体系来支撑。

And, like, that requires a pretty substantial apparatus.

其实，除非你投入了大量的时间和精力，达到能够有意义地研究那些我认为根本性的问题的水平，否则根本不可能做到。

Like, it's, like, simply not the case that you can, like, study the questions I think are fundamental unless you've invested a huge amount of time and effort kind of getting to the point where you can meaningfully do so.

对。

Mhmm.

而且，即使是在前沿做研究的那一小群人，他们的工作也依赖于一个庞大的体系。

And then moreover, that you know, even the people like, you know, you have the small group of people doing, like, fact, the research map on the frontier or whatever.

这个体系依赖于成千上万、甚至数以百万计的人在学习数学思维。

That relies on, like, a huge apparatus of, like, you know, thousands and millions or billions of people who are trying to learn to think mathematically.

你需要整个数学界来支持那一小群在前沿工作的人。

Like, you need a an entire mathematical community to support a small group of people who are on the frontier.

没有这个人才输送管道，前沿研究根本不可能存在。

Like, you just without the pipeline, the pipeline doesn't exist.

所以，如果你认为培养能够有意义地参与前沿数学的人力资本很重要，那么你仍然需要激励那些人真正投入时间和精力，达到能够参与的水平，然后以高质量、有意义的方式去做。

So if you think that's important, like, development of human capital that can, like, meaningfully engage with frontier mathematics, I think that, you know, you still have to incentivize those people to actually invest their time and effort getting to the point where they can engage and then do so in a in a high quality meaningful way.

所以现在，我认为现有的数学研究激励机制并不鼓励人们这样做。

So right now, I think, like, the existing incentive structures for math research do not do not encourage people to do that.

所以，你知道，现在如果你是个博士后，想在社区适应之前找份工作，可能就这几年。

So, you know, right now, if you're, like, a postdoc on the market, you wanna get a job maybe for the next couple years before the community adapts.

最好的办法就是多出论文，也许能证明一些旧的研究方向之类的，你可以通过不断尝试来做到这一点，直到

The best way to do that is, like, you wanna produce a lot of papers, which maybe prove, you know, old trajectories or whatever, and you can do that by playing the slot machine until

对。

Yeah.

模型能给出一个可能正确的证明，证明这样的结果。

The model produces a hopefully correct proof of such result.

所以你甚至不需要提前选定定理。

So you don't even have to pick the the the the theorem in advance.

所以，这里有个你可以做的实验。

So, like, here's an experiment you can do.

你可以拿编码器。

You can take codecs.

你可以说，上网找五个代数几何领域的最新猜想，然后证明它们。

You can say, go online and find five recent conjectures in algebraic geometry and prove them.

而且，好吧，我做过这个实验，经过一些来回折腾，我大概在一个小时内就能弄出三篇相当糟糕的论文。

And, okay, I've run this experiment and with some back and forth, I was able to, you know, in an hour, get, like, three, you know, quite bad papers.

嗯。

Mhmm.

但论文是正确的，现在它们就躺在我硬盘上，等着我发给相关的人，但把时间花在这些结果上对我来说并不划算。

But correct papers, which, okay, are now sitting on my hard drive waiting for me to email the relevant people, but this is not a good use of my time to invest in results.

对。

Yeah.

但确实，你肯定能看到有人这么干。

But, yeah, you you definitely see people doing this.

所以，你知道，在arXiv上这类东西的数量激增了。

So there's, you know, been a huge uptick in post archive.

好吧。

Okay.

大多数没什么意思。

Mostly not very interesting.

有些内容确实有意思，但你知道，很多明显质量不高，即使结果放在一年前可能会得到很高的评价，但看起来根本没有人真正参与其中。

Some of it is interesting, But I'll you know, a lot of it is kind of clearly low quality even if insofar as the like, even if the result is something that would have been, like, kind of highly rewarded a year ago, just like the there's sort of no evidence that a human being is actually engaged with it.

就像，没有人力资本或理解的发展。

Like, there's no development of human capital or understanding.

比如，有时候，你知道，我们见过一些例子，三四五篇论文在几天内相继出现，证明完全相同的定理，这明显是有人在玩老虎机。

Like, sometimes, you know, we've seen examples where, like, three or four or five papers with the exact same proof of the exact same theorem out in within a couple days of each other, which is clearly, you know, some situation where someone's playing the slot machine.

对。

Yeah.

ChatGPT 总是能找到同样的东西。

ChatGPT is kind of consistently finding the same thing.

有意思。

Interesting.

这就像在某些推理路径上发生了模式崩溃。

It's like not it's kinda mood collapse on, like, certain pads of reasoning.

对。

Yeah.

对。

Yeah.

而且我觉得，这个问题不一定随着模型变好就消失了。

And I think it's, like, not obvious that this problem goes away as the models get better.

也许会，也许不会。

Like, maybe it does, but maybe it doesn't.

就像我之前说的，数学的很多进步来自于让各种不同的想法百花齐放，人们各自去探索。

Like, I think a lot of what I was saying earlier is that a lot of progress in mathematics comes from, like, letting, you know, a thousand different flowers bloom and people Yep.

大家追随自己的好奇心，然后知识的边界以一种比较均匀的方式扩展，看着这个高维的数学空间。

Pursue their own curiosity, and then, you know, the boundaries of knowledge expand in some kind of fairly, hopefully, fairly uniform way and look at this really high dimensional space of mathematics.

然后，有时候会突然得到一些应用，或者找到老问题的答案。

And then, like, opportunistically, you you suddenly get some applications or, like, answers to old questions we found.

我不太确定，如果我们让数学探索服从于模型的兴趣，那得到的是不是一个数学家复制了一千遍，还是真的有一百万个不同的数学家在各自做不同的事情。

And it's, like, not clear to me that if if, you know, we kind of subordinate mathematical exploration to what the model wanna pursue, like, if what you're getting is, like, one mathematician duplicated a thousand times or, like, actually, you know, a million different mathematicians doing a million different things.

对。

Yeah.

我觉得，其实这挺——我是说，你知道，除了数学得通过改变激励机制来适应之外。

I think, actually, that's pretty I mean, you know, aside from, like, okay, math has to adapt by changing incentive structures.

我觉得这其实挺危险的，或者说实验室们得注意这一点。

I think this is actually a pretty, maybe, dangerous or something that the labs have to pay attention to.

因为，你知道，一方面，对他们来说成功的地方在于，这种涌现出来的推理能力显然非常强大。

Because, you know, on one side, what's been successful for them is that this emergent reasoning capability is obviously incredibly powerful.

而且我们一直在——我是说，这确实制造了很多很好的公关头条。

And we've been I mean, it's just created a lot of, like, great PR headlines.

但是，嗯。

But Mhmm.

你知道，说到你的观点，而且也是我感兴趣的地方，就是人类数学家们，他们来自各种奇怪的直觉，你懂吗？

You know, to your point and also, like, where my interests tend to is that, like, human mathematicians, they are coming from all sorts of weird intuitions that, like you know?

这就是为什么你最终会发展出，你知道，像你说的，一个如此多样化的前沿，你才能真正从中汲取，然后连接起来，产生更多的东西。

And that is why you end up developing, you know, to your point, the frontier that is so diverse that you actually can draw from and then be able to connect and produce a lot more.

所以，如果实验室们推出的这些证明大多数都收敛到非常相似的东西上，因为它们在技术上都是基于同一批文献，而那是它们目前擅长的领域，那么这确实是个问题。

And so it if it's true that most of these, like, proofs that are being pushed out by the labs are converging on very similar things because they are technically drawing on the same body of literature, and that's where they're strong at now.

这还不清楚，我是说，首先，他们的直觉是从哪里来的呢？

It's it's not clear that I mean, one, you know, where are they developing their intuitions?

是从实践中的数学家那里来的。

It's from practicing mathematicians.

还不清楚那种更强大的、多样化的直觉可能从哪里冒出来。

It's not clear where that other emergent, you know, stronger, diverse intuition might be coming from.

它可能会出现，但我对它的来源没有好的理论。

It might come, but I don't have a good theory for where it comes from.

而且也不清楚，仅仅靠测试时的计算和各种训练后的扩展，你甚至能不能——关于能否引入新能力，这本身就是一个巨大的争论。

And it's also just not clear that, you know, just with, like, test time compute and various post training scaling that you can even I mean, it's a huge debate whether you could introduce new capabilities.

而且我觉得，这恰恰应该在数学的背景下研究，因为它在哪些方面擅长、哪些方面失败，以及人类数学家擅长什么，这正好是一个非常精确的研究问题。

And it's like precisely, I think you should be studied in the math context because, like, where it's good at and where it fails, and where human mathematicians are good or is exactly a very kind of, like, precise question to study it on.

所以，这说起来很长，但对我来说，如果我们不激励足够多的人去与它互动，也许不像其他领域，它可能不会在没有人类帮助的情况下继续产生更优的结果。

So it's like a long way of saying that I it isn't clear to me that we'll maybe produce if we don't incentivize enough people to interact with it, maybe unlike other domains, it it might not actually continue producing a superior results without a human aid.

对。

Yeah.

让我再进一步说说。

Let me actually push push a bit further even.

比如说，假设模型真的变得超级厉害，甚至我们都没有加入有意义的认知多样性。

So, like, let's suppose the models become, like, really robustly superhuman, like, even, like, we're not even adding, like, meaningful cognitive diversity.

好的。

Okay.

所以我说，其实我们还是赢了。

So I claim, like, still, actually, we we still won.

好的。

Okay.

太好了。

Great.

数学家。

Mathematician.

对。

Yeah.

对。

So right.

那我们为什么？

So we why?

因为，你知道，这涉及到我们如何设计社会的问题。

So, like, it's because the you know, there's, like, a it's like there's a question on how we've we're designing society.

对吧？

Right?

对。

Like Mhmm.

也许最优的情况是，让模型去做各种数学研究，然后以一种非常不均匀、多样化的方式推进。

Maybe the optimal situation is, like, you have the models doing all sorts of math research, and, like, that leads to very in, you know, in this, highly, you know, whatever, nonuniform diverse way.

也许你不需要人类来增加那种多样性。

Like, maybe you you don't need humans to add that kind of diversity.

所以，也许那才是最优的做法。

So, like, maybe that's the optimal thing to do.

而且，最终这会带来很多应用和更深的理解等等。

And, like, in the end, that leads to lots of applications and lots of improved understanding and so on.

就是说，即使某件事是最优的，也不意味着你就会去做。

It's like, despite like, something being optimal doesn't mean you do it.

对。

Mhmm.

对吧？

Right?

就是说，没有理由认为，如果我们把数学研究的控制权交给模型，让它们自己随便搞，它们就会去做最优的事情。

Like, there's no reason to think that, you know, if we hand over control of whatever math research to the models and just let them do their own thing, that it will do the optimal thing.

而且事实上，如果我们把想要它们做的事情工具化，比如我们说，哦，让我们的生活更好之类的，嗯。

And in in fact, like, if you, you know, if we are if we kind of instrumentalize what we want them to do, like, we wanna say, like, oh, you know, make our life better or whatever Mhmm.

可能它们决定去做的，并不一定是去追求各种有趣的研究。

It might not be the case, like, be that that, like, what they decide to do that is, like, is is pursue a wide variety of interesting research.

对吧？

Right?

他们可能只会选择走捷径。

Like, they may just try to take the direct path.

我们不知道会发生什么。

We don't know what's gonna happen.

所以，我不知道，如果你相信这种广泛的、基础性的研究有任何价值的话——我确实相信，我觉得这是人类或者模型能做的最有价值的事情之一。

So, I don't know, if you believe that there is any value in sort of this broad based, like, fundamental research, which I do, look, I think that's one of the most valuable things human humans or or whatever or the models can be doing.

对。

Yeah.

我觉得，要确保这件事发生，最简单的办法就是保持一个兴趣广泛的社区。

Like, I think the easiest way to guarantee it happens is, like, to keep a community with, like, broad interests.

我们正在推动模型去做这件事，帮助我们设计一个社会，让我们朝着这个方向努力。

We're, like, pushing the models to do it, like, helping us to design a society where, like, that's what we're what we're pushing for.

对。

Yeah.

我觉得，至少我对于未来的美好愿景是，人类不会被完全剥夺权力。

Like, I think at least, you know, my my hopeful vision for the future is that, like, humans are not, like, totally disempowered.

就是说，我们对未来的方向还是有一定掌控力的。

Like, we have some control over where we're going.

而且如果是这样的话，我们最终做的事情，其实是由人类的兴趣来决定的。

And, like, if that's the case, like, what we end up doing is gonna be driven by human interests.

所以你得有那些兴趣广泛的人，而且他们还得有能力去真正追求这些兴趣。

And so you wanna have people who have lots of different interests and also the capabilities to actually pursue them.

就是说，你需要那些聪明、投入的人，而且——对。

Like, you want people who are, like, smart and engaged and, like Yep.

你知道，他们不仅能做数学思维，还能做各种类型的思考。

You know, well, can do not just, like, mathematical thinking, but all sorts of thinking.

对。

Yeah.

对。

Yeah.

我的意思是，对，一个很大的担忧是，随着AI的发展，我们可能不会开发出合适的人机交互界面，来真正鼓励我们继续保持好的思考能力。

I mean, yeah, a a great fear is, like, as AI advances, we don't develop the right ergonomic, kind of interfaces to actually encourage us to also be good continue to be good thinkers.

而且我们很容易就放弃这种思考，因为你可以直接交给AI。

And and it's so easy to kind of relinquish that because you just off.

而且那些模型在最高层的思维结构上其实并不擅长。

And and the models aren't even good at that level of, like, you know, thinking where it's like the top level structure.

但尽管如此，我们还是很轻易就依赖它们。尤其是数学，我有个自私的理由：如果你深入研究数学，我觉得这其实是一个很好的教学借口，能让你真正严谨地思考各种问题。

But despite that, it's it's so easy to and so especially for math, I mean, just like, you know, another selfish reason is if you kind of math max, you know, I think it's actually a great pedagogical excuse to actually get just really rigorous at thinking about various things.

当然，这不是数学家做数学的原因。

I mean, this is not why mathematicians do it.

但作为一个只是……

But as just somebody who's

这是我们工作的一部分。

like part of what we do.

好的。

Okay.

很好。

Great.

因为我觉得它确实帮我建立了一个很好的思考框架，不只是数学方面。

Because I it was kind of you know, I I thought it just really helped give me a very good framework to think about many things, not just mathematics.

而且作为现在一个两岁孩子的家长，我也经常在想这个问题。

And as somebody who's also, you know, now a parent of a two year old, I kinda think about this, a lot too.

你知道吗？

You know?

这不仅仅是死磕，虽然，怎么说呢。

It's it's not about kind of grinding even though, you know, whatever.

死磕还是有好处的，但也不是说要把死磕贬得一文不值。

It's still good, but, like, not to not to, shit on grinding too much.

但是，你知道，我以前和很多匈牙利数学家合作过，其中就有。

But, you know, just hearing I I used to collaborate a bunch with some Hungarian mathematicians, among them.

我听说在布达佩斯，他们小学就开始教群论了。

And I just it hurt that in in Budapest, they would just teach group theory when you're in primary school.

我就觉得，我们真应该也这么做。

And I'm like, well, we should definitely do that.

我们应该继续这样做。

We should continue doing that.

现在AI在解释方面已经很不错了，而且更容易接触到，我们其实应该更广泛地推广这一点。

Now that AI is so good at, you know, some somewhat good at explaining, but it's far more accessible, we should actually, you know, probably proliferate, that even more.

所以也许这能帮助更多人接触到前沿，而不是仅仅……

And so maybe that helps with bringing more people to the frontier rather than, you know, just

对。

Yeah.

我的意思是，这也是我担心的事情。

I mean, this is something I'm concerned about.

对吧？

Right?

不过，当然，我主要谈论数学，因为那是我最熟悉的领域。

But, I mean, of course, I mostly talk about math because that's, like, where I live.

但是，我觉得，思考这个问题的一个好处是，我们是首批受到高质量模型显著影响的职业之一。

But, like, I think, you know, one nice thing about thinking about this is that we're one of the first professions to to kind of, you know, have a significant impact of high quality models.

虽然我觉得我们可能是第一个这么公开地经历这种变化的职业，但我觉得还有很多其他职业也在经历类似的事情。

Although, I think maybe we're one of the first professions for it to happen so publicly, but, you know, my sense is that there are plenty of third professions that are

编程。

Coding.

可能吧。

Probably.

对。

Yeah.

百分之百。

A 100.

编程。

Coding.

而且不仅仅是编程，我觉得任何在电脑上做的事情，现在可能很大一部分都是由模型来完成的。

But also just like I mean, I think that, you know, anything you do at a computer, like, probably a huge amount is being done by the models at this point.

是的。

Yeah.

而且，好像也没有引起什么公开的反思。

And then, like, there's nothing a public reckoning about it.

但是，是的。

But Yeah.

你知道，因为数学能力对实验室来说，我觉得比其他人更愿意公开谈论。

You know, because, you know, math capabilities are are useful for the company for the labs to talk about, I think, more publicly than everyone else.

但是，是的。

But yeah.

而且我觉得，在这个领域保持人力资本的一个原因就是，它是所有职业的一个典范。

And I think, you know, one reason to try to maintain, like, human capital in this area is just like it's a model for all professions.

比如，我们大概还是希望有人能够有意义地参与世界，成为专家，拥有天赋和训练有素的技能等等。

Like, presumably, we still want people who are, like, meaningfully engaging with the world and, like, experts and, you know, have, like, talents and trained skills and so on.

对，就是这样。

Like yeah.

所以，我刚才说，是的。

So I was yeah.

其实数学专业和教育联系这么紧密，真的很方便，因为我觉得我们也看到了一些挑战，尤其是在大学和，嗯，中等教育阶段，AI也带来了影响。

It's actually very convenient that the math profession is so entwined with education here because, like, I think we're also seeing, like, you know, some amount of challenges, you know, among college and and Mhmm.

比如，中等教育也受到了AI的冲击。

Like, secondary education coming from AI too.

所以就像你说的，它也是一个很棒的学习工具。

So as you said, I mean, it's also an amazing tool to learn.

是的。

So Yeah.

你知道，我听说有人开始谈论他们课堂上的双峰分布，有些人真的在想办法利用AI，而另一些人只是让AI帮他们做作业，然后其他考试一塌糊涂。

As you know, people have have I I've heard heard people start talking about a bimodal distribution in their classes where there's some people who are really, like, figuring out how to take advantage of neutrals and other people who are just, like, letting them do their homework and then bombing everything else.

不幸的是，我觉得适应速度不够快，但我们肯定希望生活在一个能培养出更好思考者的世界里。

Unfortunately, I don't think that adapts fast enough, but it's like we definitely wanna be living in a world where we're producing better thinkers.

我觉得就是这样。

I think it's just Yeah.

你知道，即使我们只谈纯粹的优化游戏，我觉得那对我们来说也更好。

You know, even if we talk about just the pure kind of optimization game, I think that's better for us.

所以，对。

And so Right.

但作为一个人，我觉得如果AI崛起的同时我们反而变得更不会思考，那也太不方便了，而且这种情况很容易发生。

But as, you know, just like a human being, I'm like, that would be pretty inconvenient if we became worse thinkers just as AI ascends, and it's too easy to let that happen.

所以我们应该认真思考如何真正利用这一点，让它为我们自己的进步服务。

So we should kinda be thinking hard on how to actually take advantage of this and harness it for our own improvement

对。

Right.

也是。

As well.

是的。

Yeah.

我觉得观察这些模型挺有意思的，以它们目前的能力水平，能让你做更多事情。

I think it's like it's sort of interesting to observe, like, the models, at their current level capabilities, let you do a lot more.

它们让你能做很多以前不会做的事情，而且成本更低，低到你现在就愿意去做。

They let you do a lot of things you wouldn't have done more cheaply than, you know, cheaply enough to do them now.

但我不太确定，在很多情况下，它们是否真的提高了输出质量。

But it's not clear to me that, like, in many cases, they're actually improving the quality of outputs.

对。

Yep.

我觉得这很常见。

And I think this is common.

就像，你有一项新技术，它做的事情比以前稍微差一点，但便宜得多，所以你会突然得到很多低质量的输出，取代了之前高质量的输出。

Like, you have a new technology that's doing something a little bit worse than was previously done, but much cheaper, and so you get a lot of suddenly, a lot of, like, low quality outputs that are displacing previous high quality outputs.

但我认为，有可能以某种方式使用这些工具，实际上在各个方面都提高了质量。

But I think it's possible to use the tools in a way that actually, like, improves, you know, the quality of among along every dimension.

这只需要一些深思熟虑，以及重新设计机构来真正激励这种改进。

It just requires some thoughtfulness and some some redesign of institutions should actually incentivize that.

是的。

Yeah.

嗯，希望资本主义能在这方面发挥作用。

Well, hopefully, capitalism works there.

我确实觉得，最有价值的事情还是需要人们有效地去使用它。

I do feel like that the most high value things do require people to use it effectively.

而且现在，如果没有人类专家的参与，模型还不够好。

And and right now, the models are not good enough without, like, the human experts to actually participate.

但你说得对，绝大多数可能都是比较初级和入门级的。

But to your point, there's a vast majority of maybe, like, more junior and and entry level.

而且你知道，这些角色要适应起来也更难。

And if you know, it's it's harder for for those roles to adapt as well.

所以，如果使用AI模型的方式不对，那就会是个错误。基本上，你需要不断提升，用模型来加深你的理解。

And so the thing that would be a mistake is to use the AI models in a way that doesn't basically, you need to be ascending and using the models to deepen your understanding.

而且人性太容易懒惰了，你必须抵制这种倾向，因为那就是你开始失去的时候。

And it it's so easy for human nature just to be lazy, and you have to resist that because that is the moment that you will kind of kind of lose, basically.

所以，请原谅我用了这么激烈的语言，但事实就是，把思考交给模型真的太容易了。

And so you kind of, you know, kind of forgive the very competitive language, but it really is that, like, it's just so easy to to kind of relinquish thinking to the models.

模型其实不会思考。

The models can't really think.

所以，随着这些技术飞速发展，你持续提升这些能力，并真正利用它来增强这些能力，而不是放弃，这一点至关重要。

And so as things are ascending so fast, it's, like, critical that you continue developing those facilities and actually leverage it to to to improve those facilities rather than relinquish.

对。

Yeah.

对。

Yeah.

对。

Yeah.

我觉得，这种半专家关注度或者模型对数学问题的关注度大幅提升，带来的一件好事是，现在，你知道，好吧。

I mean, one one thing I I think has been nice about, you know, sort of this vast increase in in, like, semi expert attention or, like, model attention on math problems is, like, now there's been you know, okay.

我一直在抱怨那些灌水论文什么的。

I've been complaining about slot papers or whatever.

对。

Yeah.

比如，那些不是真正产出高质量成果的人。

Like, you know, people who are not, you you know, producing really high quality stuff.

其实，这些往往来自专业人士。

Often, that's actually coming from professionals.

我不是说，你知道，在学术数学界，有些人为了发很多论文，确实有这种激励。

Like, it's not I'm not saying, like like, you know, there are people who, you know, the like, within academic mathematics, there are incentives to produce, like, a lot of Yes.

发很多论文，这也是垃圾论文的一个来源。

A lot of stuff, and and that's, you know, that's one place to stop coming from.

所以确实也有来自非专家的垃圾论文，但我其实不觉得这是坏事。

So there's definitely also, like, stop coming from nonexperts, but that I I kind of actually don't see as a net negative.

比如说，好吧。

Like, okay.

现在网上有很多文档，你可能得翻一遍才能知道某个问题是否已经解决了。

There's a lot of now there's a lot of, like, documents on the Internet one might have to come through to figure out if a problem has been solved or not.

嗯。

Mhmm.

但对我来说，有这么多人热爱数学，这本身就是件好事。

But to me, it seems like just the fact that there's lots of people excited about math is, like, kind of a positive.

所以这挺好的。

So that's, like, a nice thing.

我完全同意。

I totally agree.

我知道我能跟你聊。

I know I get to talk to you.

而且不只是有点正面。

And not even kind of a positive.

显然，这很棒。

Obviously, it's it's great.

对。

Yeah.

对。

Yeah.

不。

No.

没错。

Exactly.

就好像突然有一束光照在上面，我可以更深入地聊数学了。

It's like suddenly there's a spotlight on it, and I can nerd out about math more.

对。

Yeah.

我其实挺好奇你有没有看到关于那个——我觉得这也是个建设性成果——秩为30的椭圆曲线的评论。

I I was actually kinda curious if you've had any comments on the I guess this is another constructive result, but the elliptic curve of rank 30.

那个昨天刚出来。

That just came out yesterday.

所以如果那个……

So if that

没错。

That's right.

是的。

Yeah.

对。

Yeah.

不是，我是说，我们还没有任何关于它的细节。

Not I mean, we don't have any details about it yet.

我知道。

I know.

有传言说……

There's rumors that

是的。

it's Yeah.

我们不知道它是怎么……我是说，这似乎是Claude Fable提出的，由……对。

We don't know how it how it I mean, so it was it's due to, I guess, Claude Fable, prompted by and Right.

一个合作者，可惜我忘了他的名字。

A collider whose name I unfortunately forget.

也许你可以在后期加上。

Maybe you can So we'll have in post.

对。

Yeah.

我只知道那个推特账号。

I just know the Twitter handle.

对。

Yeah.

对。

Yeah.

我们可以

We can

所以，是的。

So yeah.

所以，最近很多不错的结果都是在自主性不明确的情况下取得的。

And so a lot of these nice recent results have have come from with unclear amounts of autonomy.

嗯。

Mhmm.

所以我的感觉是，其中一些是半自主的，而不是完全自主的。

So my sense is that some of them are semi autonomous rather than fully autonomous.

对。

Yeah.

这样的话，我们对方法一无所知。

With this, we don't know anything about the methods.

所以，是的，这是一个有趣的构造。

So, yeah, this is a fun construction.

在不了解方法的情况下，很难说它有多重要。

Without knowing about the methods, it's very hard to say how significant it is.

对。

Yeah.

我想说的是，如果你想从历史的角度理解这类结果是如何被证明的，或者抱歉，是如何被学界理解的，它们就是挺酷的，但我觉得算不上什么大事。

What I would say is that if you want to understand kind of historically such results have been proved or sorry, have been have been understood by the community, they're, like, cool, but I wouldn't say they're, like, a big deal.

所以，这类结果通常可能会出现在某个人的个人网站上作为记录。

So, like, the typical place a result like this might go is, like, someone's website of records.

嗯。

Mhmm.

但这不算是一个分析层面的结果。

But not It's on an analyst level result.

对。

Yeah.

这不是分析层面的。

It's not an analyst.

不过话说回来，这挺酷的，而且你知道，确实有一些非常非常有才华的数学家喜欢这类问题。

But but that said, it's cool, and it's like a may you know, it definitely there were a few very, very or are a few very, very talented mathematicians who like these kinds of questions.

所以现在马尔基斯就是，对。

So now Malkis being Yep.

对。

Yep.

也许是最著名的例子。

Maybe the most famous example.

所以阿尔基斯他们一直在推动这个纪录，最近他们找到了一个秩为29的例子。对。

So Alkis and are the ones who kind of have been pushing this record for a while, and they recently found a rank 29 Yeah.

例子。

Example.

又提升了一个。

One up.

之前那个纪录就是这个。

Which was, you know, the the previous record.

对。

Yeah.

对。

Yeah.

你知道吗？

You know?

所以那些是数学家，我觉得大家挺喜欢的，现在模型能做这类事情真的很酷。

And so those are mathematicians, and I think, you know, people like it, and it's cool now that the models can do this sort of thing.

不过，是的，这非常，你知道，我总说一个模型的结果，你只能事后才能评价它。

But, yeah, it's it's very you know, one thing I always say about a model result is, like, you cannot evaluate it except in retrospect.

而且，人类数学也是这样。

And, like, this is also true of human mathematics.

对。

Like Yeah.

有时候，我们觉得某个问题很重要，或者需要非常巧妙、深刻的新想法，但实际上并不需要；有时候又确实需要。

Sometimes a problem we thought was really important or would require really neat, deep new ideas does not, and sometimes it does.

你没法提前知道。

And you can't really know in advance.

所以，当这样一个问题被解决时，总是令人兴奋的，但要弄清楚它有多重要，你还是得看看，嗯，还有Claude，我觉得Eva Howell，也许她是合作者，好吧。

So it's always exciting when a problem like that gets solved, but then, like, to figure out how significant it is, you still have to look and, like, well, and Claude, and I think Eva Howell, maybe she's the collaborator Okay.

他们还没有告诉我们他们是怎么做到的。

Have not yet told us how they did it.

所以他们有没有……

So Have they

他们是不是更保密一些？

been mostly more secretive?

我觉得他们发布了一些相关的痕迹。

I I think they released some traces for stuff.

但是

But

对。

Yeah.

所以对于这个，我觉得他们还没有发布，除非我错过了。

So for this one, I think they haven't yet unless I missed it.

对。

Yeah.

他们，他们，你知道，Levent喜欢在推特上发布他的结果。

They they they well, you know, Levent likes to tweet out his his results.

但是，是的，他一直在慢慢发布一些PDF格式的书面总结。

But, yeah, he has been, you know, slowly releasing some kind of PDF read write ups too.

我觉得他就是，你知道，在网上玩得开心而已。

I think it's you know, he's just having fun on on a fun on the Internet.

对。

Yeah.

对。

Yeah.

这让我想起你之前说的，你得评估结果是怎么来的。

It's it reminds me of when you're saying, you know, you have to evaluate how the results came.

所以我最近请了OpenAI的Mark和Metab Swanee来，他们说最让人惊喜或开心的就是证明相对简短。

So I had Mark and Metab Swanee on from OpenAI recently, and they were saying how, you know, what's kind of been the most charming or delightful is just that the proofs have been relatively short.

不是那种两百页的。

They're not, like, 200 page.

而且，你知道，这可能也对应了你说的那个磨人的点。

And, you know, maybe corresponding to your grinding point as well.

但也许这是优化过的，事后看来是选了个短的，或者你有没有发现，平均来说，你扔给GPT的东西，比如Saul或Fable，往往更短，更容易让人看懂，而不是可能直接失控，然后硬磨出来。

But maybe this is kind of optimized for in retrospect, it was picked that it was short or maybe do you find that on average, stuff that you throw at GPT, you know, Saul or Fable tends to be shorter and more kind of legible to the human or versus it might just go haywire and just grind it out.

对。

Yeah.

所以，我觉得有时候能给出简短的证明是挺好的。

So so, I mean, I I I think it is nice that we'll sometimes produce short cover proofs.

对。

Yeah.

那当然，大家都喜欢简短的证明。

So that's of course, everyone likes a short cover proof.

我觉得它们之所以没有给出又长又复杂的证明，是因为它们做不到。

I think my sense is that the reason they're not producing long, complicated proofs is that they cannot.

就像，

Like,

它们还没有能力去验证正确性。

the just like the the ability to check correctness is not yet there.

所以，你知道吗，其实如果你让模型给出简短证明，你通常可以再问它们，这个证明对吗？

So, you know, I mean, even actually, you know, if you ask the models to produce short proof, you you can then often ask them to you also just ask, is that correct?

而且他们经常会说不对。

And they will often say no.

所以，他们比六个月前可靠多了，这是肯定的，但还是会时不时出错，而且他们自己也知道错了。

So, like, they're much more reliable than they were six months ago for sure, but, like, it's still you know, they will still sometimes produce things that are just wrong, and they they know they're wrong.

对。

Yeah.

对。

Yeah.

我觉得问题在于，如果生成的东西很长，他们可能就不知道自己错了。

I think the problem with producing a very long thing is they might not know they're wrong.

所以我在想，OpenAI和Anthropic内部肯定已经解决了很多问题，只是没放出来。

And so but I wonder if, you know, presumably, internally, OpenAI and and Anthropic have probably solved a lot more problems than they've released.

嗯。

Mhmm.

而且我猜其中不少问题，他们自己都不确定答案是不是对的。

And I imagine quite a few of them are they're just not sure if they're true.

嗯。

Mhmm.

比如说，你知道，OpenAI最近发布的这10个问题清单，它们都是用Lean形式化验证过的。

So for example, you know, with these this recent list of 10 problems released by OpenAI, those were all formalized in lean Mhmm.

这当然是很强的证据，说明它们是正确的。

Which is, of course, a very good good evidence that they're true.

对。

Yep.

我毫不怀疑还有很多问题是他们没法用Lean形式化的，比如，可能前置条件还没被纳入主流流程之类的。

I have no doubt that there were a lot more that they could not formalize in lean, like, maybe it's the prerequisite options have not been put into mass flow yet, for example.

可能还有一些更长的，这也让验证变得困难。

There were probably more that were longer, which also makes it challenging to check.

对。

Yep.

所以，是的，这就是我的猜测。

So, yeah, this is my guess.