AI 真的“解决”了数学问题吗？（让我们深入探讨）| AI 现实检验

上周，OpenAI 发布了一篇新闻稿，标题是《一个 OpenAI 模型推翻了离散几何中的一个核心猜想》。

Last week, OpenAI published a press release titled An OpenAI Model Has Disproved a Central Conjecture in Discrete Geometry.

他们具体讨论的是平面单位距离问题，这个问题是保罗·埃尔德什在1946年提出的。

They were talking specifically about the planar unit distance problem, which was posed by Paul Erdos in 1946.

其实，这个问题本身表述起来相当简单。

Now this is actually a a pretty simple problem to state.

简单说就是：在一个平面上，给定一组点，最多能有多少对点之间的距离恰好是单位长度？

It basically says, what is the maximum number of pairs of points in a set of endpoints in a flat plane that can be exactly one unit of distance apart?

上世纪四十年代，埃尔德什对这个问题给出了一个答案。

Now back in the nineteen forties, Erdos proposed an answer to this question.

他没能证明，但他觉得自己知道正确答案。

He couldn't prove it, but he thought he knew what the answer was.

上周，OpenAI 宣布他们用一个大语言模型证明，埃尔德什的答案其实是错的。

Last week, OpenAI essentially announced that they had used an LLM to prove that Erdos proposed answer was in fact incorrect.

OpenAI 的新闻稿还配了一个视频，背景音乐很紧张，几位研究人员在一块小得滑稽的黑板上认真演算，解释为什么这件事非同小可。

The OpenAI press release was accompanied by a video that featured dramatic music and a group of researchers writing earnestly on a comically small blackboard as they explained why this was a big deal.

来，我们先放一段那个视频。

Here, let's play a clip of that video.

这是人工智能带来的首个数学突破。

This is the first mathematical breakthrough due to an AI.

这被认为是组合几何中最著名的问题。

It's it's been described as the the most well known problem in combinatorial geometry.

对于整个数学分支来说，这可能是最广为人知的问题了。

So for for a whole subfield of mathematics, it's like maybe the best known problem there is.

主流媒体很快对此事表现出极大兴趣。

The mainstream press soon picked up on this story with enthusiasm.

这是《新科学家》的标题。

Here's the new scientist headline.

数学家们被AI在数学领域最大的突破震惊了。

Mathematicians stunned by AI's biggest breakthrough in mathematics.

在X平台上，网友们更是炸开了锅。

People on x predictably went even more wild.

彼得·戴曼迪斯发了这样一条推文。

PeterDiamandis tweeted the following.

一个OpenAI模型刚刚证明了保罗·埃尔德什提出的一个八十年前的数学猜想，他是历史上最高产的数学家之一。

An OpenAI model just proved an eighty year old math conjecture from Paul Erdos, one of the most prolific mathematicians in history.

我们终将解决所有问题。

We're going to solve everything.

那么，这里到底发生了什么？

Alright, so what's actually going on here?

AI是不是已经达到天才水平了？

Did AI just reach genius level?

数学这门学科是不是已经被自动化了？

Has math as a discipline just been automated?

我本人是理论计算机科学家，过去发表过不少应用数学的研究成果，而且我引以为豪地拥有一个埃尔德什数为三的记录——如果你不知道这是什么意思，可以去查一下——所以，我自然对这些问题特别关注。

As a theoretical computer scientist myself who has published a lot of applied mathematics research in my days and someone who proudly boast an Erdos number of three, which you can look up if you don't know what that means, I am, for obvious reasons, particularly interested in these questions.

今天是星期四，意味着该做本期节目的AI现实检验了，这正是寻找答案的好时机。

Well, it's Thursday, which means it's time for an AI reality check episode of this show, which is the perfect opportunity to seek some answers.

那我们就照这个方向来深入探讨。

So that's exactly what we're gonna do.

我是卡尔·纽波特，欢迎收听《深度问题》，一档为在纷扰世界中追求深度的人打造的节目。

As always, I'm Cal Newport, and this is deep questions, The show for people seeking depth in a distracted world.

好的。

Alright.

我们得先弄清楚OpenAI到底做了什么，然后才能进一步讨论这对其他人意味着什么。

So we need to start by getting more specific about what exactly OpenAI actually did, and then we can get into the implications of what that means for the rest of us.

好的。

Alright.

我们今天要讨论的是单位距离平面猜想。

So we're looking at this this unit distance planar unit distance conjecture.

爱多士坚信自己已经找到了这个问题的答案。

Erdos was convinced that he had identified the answer to the question.

我这儿不想讲得太专业，但简单说一下，爱多士认为，如果你在平面上放置n个点，那么其中最多有多少对点之间的距离恰好为1，这个数量不会超过n的(1 + c/双对数n)次方。

I I don't wanna get too mathy here, but just to say it quickly, Erdos thought that if you were placing n points into the plane, the maximum number of points that you could get to be a unit distance apart would upper bounded by n raised to the power of one plus some constant c divided by the double log of n.

你现在听我说的时候，大概也注意到了，这个求和式里的第二项，随着n越来越大，会趋近于零。

Now as you're probably noticing as you listen to me, that second term in the sum is going to tend towards zero as n increases asymptotically speaking.

所以从渐近的角度来看，这个结果会越来越接近纯粹的线性关系。

So this result, the answer asymptotically is going to approach plain linear as the point set increases.

这个答案真是简洁漂亮。

That's a really elegant answer.

埃爾德什坚信这个结论是对的。

Erdos was convinced that was right.

很多其他数学家也默认这是对的，毕竟埃爾德什通常都是对的。

A lot of other mathematicians just assumed that that was right because Erdos is usually right.

于是大家花了很长时间，试图证明这确实是根本的上限。

And so people tried for a long time to prove that was indeed the fundamental limit.

而OpenAI的做法是，发布了一篇论文，说：不对，这个结论是错的。

Now what OpenAI did was they released a paper that said, no, that's wrong.

我们找到了一个反例。

We actually have a counterexample.

我们有一种安排点的方法，能让点的数量更多。

We have a way of placing points that is going to be have more points.

即使n越来越大，我们也能让单位距离的点对数量超过那个上限。

We're gonna feature more points at unit distance than that limit even as n increases.

我觉得真正的上限大概是n加一，再加上某个固定的小常数ε，这个ε不会随着n增大而趋近于零。

I believe the actual bound is something like n plus one plus some small fixed constant epsilon that stays fixed as you increase n as opposed to approaching zero.

他们确实找到了一个反例。

So they they had a counterexample.

但他们并没有说，这才是正确的答案。

They didn't say, here's the right answer.

也没有指出这个极限到底是什么。

Here is what the limit is.

更没说什么是最好的情况。

Here's the best.

也没说最多能有多少个点。

Here's the most possible point.

所以他们并没有用一个更好的猜想来取代阿拉多西奥的猜想。

So they didn't replace Aradocio's conjecture with a better conjecture.

他们没有证明阿拉多西奥的猜想，但提供了一个反例构造，说明他以为的正确答案根本不可能成立。

They didn't they didn't prove Aradocio's conjecture, but they provided a counterexample construction that showed the thing he thought was the right answer couldn't possibly be right.

那么，他们是怎么做到的呢？

Now how did they do this?

他们用了一个具备推理能力的大模型。

Well, they used a reasoning LLM.

也就是说，这是一种经过调优的大模型，能够像人一样出声思考，让思路自由发散。

So that's an LLM that has been tuned to essentially talk out loud, to sort of think out loud and wander with its thoughts.

我们最早在2024年见到这类推理模型，比如O1和O模型，还有那个DeepSQL的推理模型。

We we first saw the first reasoning models back in 2024 with, you know, o one and the o model, a deep SQL reasoning model as well.

本质上，推理模型是把原本静态、没有记忆的大模型，变成能模拟动态计算和记忆的过程——因为它在生成每个新词时，都会回顾自己之前说过的所有内容。

Basically, reasoning models are a way of taking an LLM which are static and have no memory and having them approximate something like more dynamic computation with memory because it can sort of as it rambles, right, it's looking at everything it said so far when it produces a new token.

所以当你在自言自语的时候，你可以利用之前所有的思考来生成下一个词。

So it can, if you're rambling, you have a you're thinking out loud, you can use all of that thinking and producing the new token.

所以这就像是有了某种记忆，这种发散思考也能带点动态性。

So it's like you have some memory and this wandering could be somewhat dynamic.

你可以在里面实现一些基本的迭代或者循环式思考，对吧？

You can some basic like iterative or looping type thinking in it, right?

他们用了一个推理模型，具体 prompting 了多少次、问了哪些问题我也不清楚，但有一次他们问到这个具体问题时，模型输出了一大段回答，一群数学专家仔细研究了这段内容。

So they use a reasoning model and what they did is I don't know how many times they prompted it or on what questions they prompted it, but on one of the times they prompted it about this particular problem, The model spit out a very long transcript of an answer and a team of expert mathematicians poured over this answer.

在这段长长的思考过程里，他们发现了那个成为反例的核心想法。

And in this long chain of thought transcript, they identified in there the core idea that became the counterexample.

这些数学家并不是直接从这段文字里挖出了反例的灵感。

So these these mathematicians didn't pull that counterexample idea out of this transcript.

他们是把想法打磨了一下。

They polished it.

把表达改得更规范了。

They wrote it properly.

他们进一步展开，整理成一篇简明扼要、更容易读懂的论文，这才是 OpenAI 最终发布的内容。

They elaborated it and put it into a short, concise, much more human readable paper, and that's what OpenAI actually posted.

所以这个大模型并没有直接发布那种精炼的、五页纸的论文之类的成果。

So the LLM did not post this sort of elegant, you know, five page paper, what have you.

是人类数学家做的，但他们是从这个模型在思考埃迪斯问题时生成的那段冗长思维链条中，获得了写这篇论文的灵感。

A human mathematicians did it, but they got the idea for writing this paper out of this really long chain of thought transcript that this model produced when cogitating about the Erdos problem they asked it about.

对吧？

Right?

事情就是这么发生的。

So that's what went on.

明白了。

Alright.

事情的经过就是这样。

That's what happened.

接下来我们来聊聊大家对这件事最关心的几个核心问题。

Let's now get into the core questions that people have about this thing that just happened.

第一个问题：这个结果真的重要吗？

Question number one, is this result really important?

对吧？

Right?

之前有过一些情况，AI公司声称他们解决了这些重要的数学问题，结果其他数学家出来指出，这些问题早就被解决了。

There's been some issues before where AI companies have claimed we solve these these important math problems and then other mathematicians came along and said, well, that's already been solved.

你可能只是在训练数据里见过，或者这个问题其实微不足道。

You probably just saw in your training data or that problem is really minor.

这次是同样的情况，还是这确实是个重大突破？

Is this the same or is this actually a really important problem?

答案是肯定的，这个问题确实重要。

And the answer is yes, it is important.

它重要，是因为这是一个广为人知的问题。

It is important in the sense that it's a well known problem.

一直以来，大家都默认爱多士提出的答案是正确的。

And everyone just assumed that Erdos proposed answer was correct.

所以当数学家们发现他其实错了的时候，都感到非常意外。

So it really was surprising to mathematicians to learn, oh, he was wrong.

之前从来没人想到过这个。

No one had come up with that before.

我不清楚到底有多少人在尝试证明这个反例。

I don't know how many people are actually trying to prove the counterexample.

之前从来没人想到过这个。

But no one had come up with that before.

好吧。

All right.

而且，这个成果完全有资格发表在顶级期刊上。

Also, it's a result that you could publish in a top venue.

对吧？

Right?

我猜他们肯定会这么做。

And they're going to, I assume.

不过，这里头有些复杂的问题，比如该怎么引用，作者署名又该怎么算。

I mean, I there's some complexities about how do you cite this and how do you do authorship.

对吧？

Right?

但如果是一个人独立想出这个结果，没靠大模型，那直接就能发《数学年刊》了。

But but if a human had come up, you know, if they had come up with this exact result without the help of an LLM, boom, annals mathematics.

对吧？

Right?

证明埃尔德什错了，这可是件大事。

It's a big deal to show that Erdos was wrong.

这确实是个重要的成果，让很多人感到意外。

So it's an important it is an important result, and it surprised a lot of people.

好的。

Alright.

第二个问题，很多人在问。

Question number two that a lot of people are asking.

这是否意味着大模型现在比人类数学家还聪明？

Does this mean LLMs are now smarter than human mathematicians?

简短的回答是：不是。

The short answer here is no.

但我还想给你详细解释一下。

But I wanna provide you a long answer as well.

我想说明一下，在这个例子中到底发生了什么、没发生什么。

I wanna provide some context about what is and is not actually happening in this particular example.

我想先读一段数学家托马斯·布卢姆的评论，他是研究埃尔德什开放问题的顶尖专家。

Now I wanna start here by reading some comments from the mathematician Thomas Bloom, who's a world's expert on Ardos's open problems.

OpenAI 发布了一篇配套的评论文章，收集了多位顶尖数学家对这一成果的看法，托马斯·布卢姆就是其中之一。

Now OpenAI released a companion commentary paper that collected comments from eminent mathematicians on this result and Thomas Bloom was one of the mathematicians they asked for his comments.

所以我打算念一段布卢姆在 OpenAI 发布的这篇评论文章里的原话。

So I want to read to you from Thomas Bloom's comments about this new result published in the commentary paper that OpenAI put out.

布卢姆一开始说了很多，但我接下来要念的这段话是从他这句话开始的：如果这篇论文的成果是解决了单位距离问题，那才真是了不起。

So Bloom starts by saying, I mean, this quote, he says a lot of things, but the quote I'm going to read here starts with Thomas Bloom saying, if the result of this paper was a proof of the unit distance problem, that would be truly incredible.

对吧？

Right?

所以退一步说，他的意思是，如果这个大语言模型真的找到了证明阿洛斯猜想成立的方法，那可比找出一个反例难多了，那才真叫了不起。

So just to step back here, what he was saying is that if if the LLM had found a way to prove that Ardos's conjecture was correct, that's a much harder thing than proving a counterexample that really would have been incredible.

对吧？

Right?

他接着说，虽然听到这个结果时他确实很惊讶，但后来得知这只是一个反例的构造，惊讶就减了一半；再后来了解到，这个构造本质上是对阿里多什原始格点构造的一种自然推广——尽管非常复杂——惊讶就更淡了。

He goes on to say, while I was still very surprised to hear of this result, this was dampened slightly when I learned it was a construction of a counterexample and still further when I learned that the nature of the construction being with the benefit of hindsight, a natural, albeit highly non trivial generalization of the original lattice based construction of Aridoshis.

好吧。

All right.

所以，让我再把他说的这段话翻译一下。

So again, let me translate the mass speak here.

他说，刚听说这个问题被解决的时候，他心想：天啊，这也太惊人了。

He's like, okay, when I first heard this problem was solved, I was like, oh my God, that's incredible.

但后来发现其实问题没被解决，只是找到了一个反例，推翻了之前的猜想。

And then when he learned it wasn't solved, there's a counter example to the existing proposed solution.

他的热情一下子就降温了。

His enthusiasm was dampened.

后来他发现，这个反例的构造其实并不是什么全新的数学突破，只是沿用了埃尔德什当初提出的那个构造，而那个构造本来就很接近他的答案。

And then when he saw that the actual counter example construction wasn't some new original leap of mathematics, it actually was just taking the original construction that Erdos had proposed that he thought matched his answer.

然后我就不太知道该怎么说了。

And then just, I don't know how to say that.

我说得可能不太准，但 basically 是他们把一些标准的代数嵌入方法应用到了原始构造上，做了一个虽然不简单、但也不算太意外的推广。

I'm gonna say this wrong, but basically they applied a, the LLMs of what would happen if we apply to sort of standard and they had the generalized to some was not trivial, but if we apply to standard sort of like algebraic embedding, I guess would be the word to the original solution.

结果发现，这个结构里单位距离的点比埃尔德什原先以为的还要多。

Hey, it turns out this thing has more points with unit distance than than Aridos thought.

对吧？

Right?

所以答案其实早就藏在附近了，只是没人之前找到而已。

So it was the answer was kind of in the near field around the right.

它就在那儿，离得不远，只是以前没人注意到。

It was around then it was nearby just no one had found it before.

好吧。

Alright.

那我继续了。

So let me go on.

我再念一段Bloom的内容。

I'm gonna read some more from Bloom.

仔细看看这个构造，就能更清楚为什么以前没人发现它。

On examining the construction, it becomes more clear how people had missed this before.

这需要好几个不太可能同时发生的情况：首先，一个优秀的数学家得花大量时间去思考单位距离猜想。

It requires the confluence of several different unlikely events that a good mathematician is one, spending significant time in thinking about the unit distance conjecture in the first place.

其次，尽管普遍认为这个猜想是对的，他仍认真尝试去推翻它。

Two, seriously trying to disprove it despite the off repeated belief of that it is true.

第三，他相信把原始构造推广到其他数域会有收获，并愿意投入大量时间去探索这类构造。

Three, believes that there is mileage in generalizing the original construction to other number fields and is so and so is willing to expend significant time in exploring such constructions.

第四，他还得足够熟悉类域论的相关部分，知道如何用现有理论来解决关于无限扩域、带有合适参数的数域的恰当提问。

And four, sufficiently familiar with the relevant parts of class field theory to recognize that the appropriately phrased question about infinite towers, a number of fields with appropriate parameters can be solved using existing theory.

这个AI满足了所有这些条件，它的成功也呼应了以往的成就。

The AI met all of these criteria, and its success here echoes previous achievements.

它常常通过坚持探索人类认为不值得花时间的路径，产生最出人意料的成果，这种能力结合了超凡的耐心和对大量专业技术工具的熟悉。

It often produces the most surprising results by persevering down paths that a human may have dismissed as not worth their time to explore, combining superhuman levels of patience with familiarity with a vast array of technical machinery.

明白吗？

Alright?

所以他在这里说的是，要找到这个答案，得有很多条件凑在一起，这就是为什么以前没人发现。

So he's he's explaining here a lot of things had to come together to find this answer, which is why no one had before.

一旦你看到了，其实也没那么难。

It's not that hard of an answer once you see it.

那我们之前怎么就没找到呢？

So why had we not found that?

他列出了几个原因，比如没人真正在找这个答案。

And he went through these reasons like people weren't looking for it.

大家都觉得埃尔德什是对的。

They thought Erdos was correct.

得有两个概念碰在一起才行，而且你还得两个都懂。

It was there was you had to there was like two concepts that came together and you would have to know both those concepts.

这背后确实需要极大的耐心。

There's some perseverance that was required.

但这里有句话我想特别提一下，因为它能帮我们更 broadly 看清数学界正在发生什么。

But there's a phrase in here that I want to pick up on because it's going to tell us more generally what's happening in mathematics.

布卢姆说，这个成果‘呼应了’以往的一些成就。

Bloom said, this result has quote echoes it quote echoes previous achievements in quotes.

那他这话到底是什么意思呢？

So what's he talking about here?

近年来，计算机辅助数学工具在专业数学家中的使用突然爆发式增长。

Well, in recent years, there's actually been an explosion in using computer aided math tools and professional mathematicians.

这类工具其实早就存在了，但最近的变化是，我们把它们和大语言模型结合起来，做了增强。

Now these type tools have been around for a long time, but what's happened more recently is we've combined and augmented these tools with LLMs.

大语言模型加上现有的计算机辅助工具，催生了大量新发现。

LLMs plus existing computer aided math tools has created an explosion of new results.

这些新成果往往有共同的特点：它们太繁琐了，普通人根本没耐心去深入探索。

It's it's uncovering a lot of results that tend to have the same sort of characteristics, mainly that they are too tedious for most humans to fruitfully pursue.

对吧？

Right?

这是因为这些结合了大语言模型的计算机辅助工具所处理的结果，往往需要系统性地搜索某个空间，而这种工作实在太枯燥了。

It's because the results that these computer aided tools with LLMs are looking at, they often require either like a systematic search of some type of space, which would be so boring.

连新来的研究生都不愿意做，而且得花上好几年。

You wouldn't even want like a new grad student to do this and it would take forever.

此外，这些工具还能调用海量的现有成果和方法，远超任何数学家能记住的内容。

Also, these tools can draw on lots of different like a vast knowledge of many different existing results and techniques that could be trained on it many more than most mathematicians can keep in their head.

所以它们愿意系统性地探索各种可能性，组合不同的方法，看看哪些行得通。

So it's willing to like systematically explore answers, mix and match different approaches and see what works.

很多这类系统还会使用形式化证明验证器，可以尝试大量方案，筛选出有效的结果。

Can also a lot of these systems will use a formal proof verifier so it can try a bunch of stuff and see what works.

这对数学领域影响巨大，而托马斯·布鲁姆的意思是，这个成果并不是AI突然展现出的全新能力。

So it's been really big for mathematics and this so this result, what Thomas Bloom is saying is this result is not some brand new capability that we didn't know that AI had.

他说这其实延续了过去几年我们一直在做的那些研究路径。

He says it's in the he's saying it's in the trajectory of those existing type of results we've been doing for the last years.

在AI数学工具真正擅长的这个领域，这个说法是不对的。

It's false in that sweet spot where AI enabled math tools really work well.

不过我也得公平地说。

Now I want to be fair.

OpenAI的这项成果确实有两点，跟最近这场AI辅助数学研究的热潮不一样。

There's two things about OpenAI's result that do separate it from this sort of existing recent explosion in AI augmented computers aided math work.

第一，他们至少声称，这完全是靠LLM的提示完成的，对吧？

Number one, this was done at least they claim just purely with an LLM prompt, right?

数学家们用的工具通常采用模块化架构，把好几种不同的模型连在一起。

The tools that mathematicians are using tend to be using modular architectures with many different types of models hooked together.

你会用到一个LLM，还会用到一个形式化证明验证器，通常用的是像Lean这样的形式验证语言，而LLM对这种语言很在行。

You'll have an LLM, you'll have a formal proof verifier usually using a formal verification language like lean which LLMs can speak very well.

还会有一套复杂的控制逻辑。

You'll have some sort of complicated control logic.

还会针对你研究的特定数学领域，对LLM进行专门的训练。

You'll have specialized training of the LLMs on very specific types of math techniques that are relevant to the fields you're looking at.

但这不是那种情况。

This was not that.

没有复杂的辅助结构。

There was no elaborate scaffolding.

就是给一个推理模型发了个提示，它一口气写了150页，结果在里面找到了答案。

It was actually just a prompt to a reasoning machine that just talked for like a 150 pages and in there they found an answer.

对吧？

Right?

所以这个成果的不同之处就在这儿。

So that is different about this result.

不过我得补充一点。

I wanna give a caveat here though.

我觉得他们用纯大模型来做这事，更多是营销噱头，而不是实际优势。

I think that this is really the fact that they're using a pure LLM to do this is more marketing than utility.

真正该用的方式，还是模块化架构，把大模型和计算机结合起来辅助数学研究。

Actually, the modular architecture tools are the right way to use LLMs plus computers to help with math.

用普通的大型语言模型来做这事，效率低、成本高、还特别麻烦，尤其是看他们得生成多少份转录文本，检查多少个问题，还得花大力气从这么长的啰嗦推理链条里挖出一点洞见。

It's it's it's I I would assume it's inefficient, expensive, and complicated just to use regular LLMs, especially as shown by they had to God knows how many transcripts they produced and how many problems they looked at and how much work was required to dig through this long rambling chain of thought to find an insight.

对吧？

Right?

我觉得这不是获得这种结果的正确方式。

I don't think this is the right way to get this type of result.

我觉得他们这么做，其实是为即将发布的新模型造势。

So I think they're using this to try to it's a new model they're about to release.

就像Anthropic宣称他们的Mytho模型能发现其他模型都找不到的bug，可实际上，那些bug多数其他模型也能发现。

So just like Anthropic made the claim that mythos can they can find bugs that no other model could before, even though it turns out that like actually largely they could.

这样一来，大家就会更愿意直接用他们的Mytho模型了。

So that you would be more willing to just use their mythos model in general.

我觉得这事儿是OpenAI在操作。

I think this is OpenAI.

他们会说：嘿，这模型连数学题都解出来了。

I'd be like, hey, it solved the math problem.

希望现在会有公司愿意付费使用这个模型。

Hopefully companies will now pay to use this model.

我觉得这就是背后的真实情况。

I think that's what's going on here.

我想提一下，就在OpenAI发布这个消息后不久，谷歌旗下的DeepMind也发表了一篇论文，推出了Alpha Pro Nexus，这是一种我刚才说的模块化架构系统，里面用到了专门针对数学优化过的LLM。

I want to point out for example, right after this OpenAI announcement, Google, Google DeepMind put out their own paper announcing the Alpha Pro Nexus, which is a modular architecture system of the type I'm talking about in which you have LLMs tuned on math.

还有自动证明求解器。

You have proof solvers.

以及一套非常复杂的控制逻辑。

You have a really complicated control logic.

系统里有主代理和子代理，据我理解，它们会协同工作，有条不紊地决定该给LLM发什么提示、探索哪些方向，然后把结果送进证明器验证，再反馈调整。

You have agents and sub agents that are systematically the control logic, I believe if I understand it correctly, is helping the systematically guide the types of prompts to the LLMs and what spaces to explore and then taking the answers and running through the proof solver and giving feedback.

这种全新的模块化架构，是目前最前沿的，他们刚公布说用它处理了353个开放的数学难题——和之前那类问题一样，结果成功解决了其中9个。

And this is this new, this modular architecture, the cutting edge modular architecture, they just announced they ran it on 353 open problems, open airdos problems, same type of problems this was from and it solved nine of them.

剩下的没解出来，但至少解决了9个。

The other ones are couldn't solve, but solve nine of them.

运行成本还挺低的。

And it was pretty cheap to run.

这些模型都很小。

These are small models.

不是那种两万亿参数的通用推理大模型。

These are not 20,000,000,000,000 parameter general reasoning LLMs.

所以，做数学可能就得这么搞。

So that probably is the way to do math.

这并不意味着问题本身不重要。

This doesn't discount the importance of the problem.

我只是说，纯用大语言模型这个方法是新的，但未必有多关键。

I'm just saying the fact it's a pure LLM is new, but maybe not that important.

明白了。

Okay.

OpenAI这个结果的另一个特别之处，在于它解决的是个重要问题。

The second thing that made this OpenAI result novel is the fact that it was an important problem.

这无疑是它最突出的特点——大家早就知道这个问题。

That's by far its biggest distinguishing factor is like people knew this problem.

如果你是组合几何领域的研究者，就像我了解在座许多人都一样，这可是个众所周知的问题。

If you're a combinatorial geometrist, as I know so many of you in my audience are, this was like a this was a well known problem.

所以，这是第一个被成功解决的有趣难题。

And so this is like the first cool problem that was solved.

用大语言模型配合计算机工具来寻找证明，这种做法其实已经持续一段时间了。

This technique of using LLMs and computer tools to help find proofs or just proofs.

但这次是第一个被攻克的重大问题。

This has been going on for a little while now, but this is the first like major problem that was solved.

这又为OpenAI的成就添上了一笔亮点。

So that is another like a feather to put an OpenAI's cap.

不过这里还得提个注意事项，就像Bloom强调的，这次成功可能有点运气成分——因为这个广为人知的问题，恰好存在一个相对简单的反例，而这类工具正好擅长处理这种类型的空间。

Again, there's another caveat here as Bloom was kind of emphasizing this may have been a little bit of a luck in the sense of this well known problem happened to have a relatively easy counterexample that existed in the type of space that these type of tools are really good at.

对吧？

Right?

这并不意味着我们现在就能解决所有这类开放性问题。

Not that we now can solve all these type of open problems.

就像Alpha Proof Nexus展示的那样，这是一种更高级、更自动化的应用，系统性更强，但它也只能解决我提到的353个类似问题中的9个。

As shown now by the the, you know, Alpha Proof Nexus, which like is a a probably a higher power, better application of this type of thinking, much more automated, much more systematic, and it could only solve nine of the 353 similar style of problems I pointed at.

对吧？

Right?

所以，大多数问题在某种程度上还是解决不了的。

So again, most problems for whatever reason are unsolvable.

很多问题本质上就是无解的，还有很多是因为它们的解法根本不在这类工具擅长的思维空间里。

You know, a lot of them because they're actually literally unsolvable and a lot of them because they're the types of solutions don't fall into that that style of solution space where these tools are well suited.

好吧。

Alright.

这其中确实有不少限制，但这就是目前的实际情况。

So there are some caveats to that, but that's what's going on here.

好吧。

Alright.

第三个问题。

Question number three.

这是否意味着所有同样困难的挑战现在都能被人工智能攻克了？

Does this mean all equally hard challenges will now be conquered by AI?

我在X平台上确实有这种感觉。

I definitely got that sense on x.

数学家们并没有这么说。

Mathematicians aren't saying this.

我不觉得OpenAI在暗示这一点，他们根本没这么说。

I don't think OpenAI, they're implying it, but they're not saying this.

但我觉得网上普遍的感觉是：这事儿太难了，根本做不到。

But I think the the feeling we have online is like, well, that's a really hard thing to do.

这是人类曾经尝试过却没能解决的问题。

It's a problem humans tried to solve, but couldn't.

这是否意味着我们现在拥有了超越人类的智能，能带来超凡的表现，以及各种我关心的成果？

Does that mean we have the superhuman intelligence that now can give us super intelligent performance and all sorts of things I might care about.

难道我们现在该学《终结者2》里的莎拉·康纳，单手上膛霰弹枪，顺便秀一把肱二头肌吗？

Is it time to learn like Sarah Connor in Terminator two, how to cock a shotgun with one arm and show off our awesome guns by which I mean biceps?

答案很可能也是否定的。

The answer here is likely no as well.

还记得这档播客之前那一期‘AI现实核查’里，我提到过两种看待AI能力的思维模型吗？

Remember in a previous AI reality check episode of this podcast, I said there's two mental models for thinking about AI capabilities.

一种是‘水位上升’模型。

The one model is a rising water one.

AI的能力在提升，就像水位慢慢上涨，而各种难题就像不同高度的山峰。

The capabilities grow and it's like a water level of capability rising and you have mountains that represent problems of different difficulties.

水位一涨，所有同等难度的山峰就都被淹没了。

And as the water rises, all mountains of that difficulty are now solvable.

如果水位高到能淹没埃爾德什开放问题这样的高峰，那那些更简单、跟我们日常生活相关的难题，自然也全都被解决了。

So if the water is high enough to cover the Erdos open problem peaks, then like it's covering all these other even easier problems that are relevant to our everyday life.

所以，我们马上就要迎来巨大的颠覆性影响了。

And therefore, we're about to have really big disruptive impacts.

但这种看法其实并不正确。

That however is the wrong way to think about AI development.

我觉得更准确的比喻是，AI能力像一条主河道，我们正在探索它的支流。

I said the better way to think about AI capabilities is like you have like a main river and you're exploring tributaries into the river.

每一条支流都代表一种不同类型的问题或应用场景。

And some of those so each of these tributaries is like another type of problem or application.

有些支流进展顺利，另一些却很快遇到障碍，根本无法继续前行。

And some of these tributaries are able to make a lot of progress and other ones you almost immediately get stuck and it becomes non navigable.

所以，哪怕某条支流取得了突破，也并不能说明下游的另一条支流就一定同样容易探索。

And so making progress on one tributary doesn't really tell you anything about whether this other tributary down the river is gonna be equally as explorable or not.

我一直主张这才是正确的视角，它能帮我们更准确地理解这些数学成果的意义。

That is what I've been arguing is the right model, and I think that helps us better place these math results in context.

自从ChatGPT让生成式AI和大语言模型进入大众视野以来，计算机科学家就一直说，LLM最擅长的两个领域是编程和数学推理。

And remember, ever since like ChatGPT put generative AI and large language models on people's radar, computer scientists have been saying there two areas in particular in which LLMs are gonna be very well suited, computer programming and mathematical reasoning.

这是因为这两个领域共享着四个共同特征。

And this is because those two problems share four elements.

它们处理的都是高度结构化的语言，要么是编程代码，要么是数学符号。

They deal with highly structured language, either computer code or mathematical notation.

它们都有明确的对错标准。

They have clear notions of correctness.

这个程序能编译通过吗？测试能跑过吗？

Does this program compile and pass the test?

这个证明是对的吗？

Is this proof true?

这个数学结论正确吗？

Is this math result right?

用于训练编程模型的数据几乎是无穷无尽的。

There's endless data to train on the computer programming tuned models.

我们可以用网上海量的代码示例来优化模型，比如有人提问，别人回答的那种场景。

We can tune it on the sort of endless examples of code online where people ask questions, other people give answers.

数学甚至更好，因为我们可以人工生成大量不同类型的数学题和正确证明，反复训练、不断优化。

Math is even better because you can actually artificially synthetically create data example after example of many different math problems and correct proofs and tune it again and again and again with it.

所以它在特定类型的数学推理上特别擅长。

So it's really good at particular types of mathematical reasoning.

而且不管是编程还是数学，都有专业人士愿意用这些复杂的工具，精心打磨出优质的结果。

And also in both cases, programming and math, you have expert users who are willing to use hard tools and massage good results out of it.

如果你告诉我，有个超强的AI工具，能一次性生成150页的内容。

If you told me I have this great AI tool, it's going to spit out a 150 pages.

你得一页一页地翻，试着从中拼凑出有用的部分。

You have to comb through it and try to piece together maybe in there.

也许里面藏着能解决我业务问题的洞见。

There's a useful insight for like my business problem.

普通上班族可能还在琢磨怎么找回纸夹和Word在哪里。

The average business person is going to say, I'm still trying to figure out where the paper clip and Microsoft Word went.

但数学家会说，当然没问题。

But a mathematician, oh yeah, sure.

我们来做吧。

We'll do this.

我喜欢用硬核工具。

Love hard tools.

好吧。

All right.

所以现在来看，编程和数学就像是生成式AI这条大河的两条主要支流，都已经证明了非常可行。

So you can think about programming and math is like the two major tributaries right now of the Gen AI River that have proven to be very navigable.

因此我们投入了大量资源去探索这两条路径，继续推动进展，毕竟作为一家AI公司，你总得能报告一些进展吧。

And that's where we're therefore putting a lot of our explorers and continuing to try to push progress because if you're an AI company, you want to report progress on something.

但你在这条支流上取得的进展，完全不能说明另一条支流的情况。

But being able to explore that tributary tells you nothing about the other.

所以千万别误以为，只要比解决一个AI难题简单点的事情，AI现在就能轻松搞定。

So do not think the absolute wrong mental model is anything that is easier than solving an airdos problem is something that AI can now do.

事实上，我之前就提过这一点，加里·马库斯在最近一篇专栏里还引用了我说过的话。

If anything, and I pointed this out, Gary Marcus quoted me saying this in a sub stack he put out soon after this announcement.

更讽刺的是，就在OpenAI即将上市、营收压力加大的时候，他们最得意的案例居然是帮助专业数学家生成离散几何的证明。

If anything, the fact that with a IPO looming revenue concerns mounting that the use case that OpenAI is crowing about is we are helping mathematicians, professional mathematicians on creating discrete geometry proofs.

恰恰相反，这反而证明了支流模型的正确性。

If anything, that is a huge vindication for the tributary mental model.

对吧？

Right?

很难再找到比专业数学家研究证明更不赚钱的领域了。

There could be few fields less lucrative than professional mathematicians working on proofs.

这个领域根本没人投钱，因为它不产生收益。

There is no money spent in that field because it doesn't generate money.

纯粹是为了知识本身而追求的高深学问。

It's incredibly esoteric knowledge for knowledge's sake.

如果你真想让投资人放心，说我们肯定能赚钱，那绝不会拿最没商业价值的应用来吹嘘。

And so if you really want to impress investors and say, we're gonna be okay, we're gonna make money, you wouldn't be bragging about the least lucrative possible application.

你肯定会说，看看这个应用，平均帮每家公司省了一亿美元。

You would be bragging about, look at this application, which is saving companies on average a $100,000,000.

我们刚刚创造了十亿美元的收入。

We just generated a billion dollars of revenue.

我们只是帮这些公司把运营成本砍掉一半。

We just help these companies cut their operating conditions by half.

这比电力还要重要。

This is going to be more important than electricity.

每个人都需要这个工具。

Everyone needs this tool.

你该宣传的应该是这个，而不是那个。

That's what you'd want to be announcing, not.

有个挺有意思的代数领域，如果你把一个普通的二维网格稍微调整一下，就能在单位距离内塞进远超线性数量的点。

Oh, there's an interesting algebraic field in which if you move a standard, otherwise two dimensional square lattice, you're able to squeeze something that is super linear in terms of points at unit distance.

对吧？

Right?

这种事根本不用拿来宣传。

That's not what you would announce.

如果解决这个问题真能让你拥有天才级AI，那你早就用它去干所有能赚钱的聪明事了。

If it was true that solving that problem meant that you now had genius level AI, you would use that to solve all the economically useful things you could do with an automated genius level artificial intelligent mind.

所以我认为，我们眼下专注的这件事恰恰证明了：擅长一件事，并不意味着就能把其他事也做好，或者我们已经看到了更多有经济价值的实例。

So I think the fact that this is what we're focusing on just vindicates the idea that just being good at one thing doesn't make you better at other things, or we've seen more economically productive examples.

第四个问题。

Question number four.

这对我们理解数学的未来意味着什么？

What does this tell us about the future of math?

数学的未来令人振奋。

The future of math is exciting.

我得把这一点说清楚。

Let me be really clear about this.

就像现在编程领域已经深度融入了基于大语言模型的工具一样。

Just like how computer programming now has LLM based tools deeply embedded in it.

只是我们还没完全搞明白该怎么用这些工具。

It's just it's we we haven't really figured out exactly how to use them.

换句话说，这个方向还没稳定下来。

Like, we haven't stabilized that yet.

现在还充斥着一堆乱七八糟的情况，比如代理监督代理，而这些代理又去监督监督者。

There's still a lot of nonsense of like agent supervising agents who supervise the agent supervisors.

到了一天结束的时候，我们发现它们写的代码连个‘Hello World’都跑不通。

And we look up at the end of the day and they have a broken hello world code or whatever.

我们还在摸索中，但LLM工具确实彻底改变了程序员的工作方式。

Like, you know, we're still figuring it out, but like LLM tools are completely changing how programmers work.

过去一年左右，专业数学领域也出现了类似的变化。

Something similar has been afoot in professional mathematics for about the last year or so.

计算机辅助工具一直都在，而且非常强大，但用起来太烦人了。

Computer aided tools have always been there and very powerful, but they're too annoying to work.

但一旦把LLM加进来，尤其是专门针对数学推理优化过的LLM，再加上OpenDeepMind那种高智商的模块化架构，效果就完全不同了。

But when you add LLMs into the picture, especially LLMs that are tuned for doing this type of mathematical reasoning, especially when you're in the very smart type of modular architectures that OpenDeepMind is producing, it really helps.

我本人没做过专业应用数学方面的研究。

Now I haven't done I haven't published professional applied mathematics research.

这纯粹是我的感觉。

That was my feel.

我是一名分布式算法理论家，专注于分布式算法领域的应用数学。

I'm a distributed algorithm theorist, an applied mathematician within distributed algorithms theory.

几年前，我暂停了数学研究，转而去帮助创建乔治城大学的数字伦理中心，并更多地投身于面向公众的技术批判工作。

A few years ago, I stopped doing mathematical research to help start up the center for digital ethics at Georgetown and do a lot more of this public facing technology criticism.

但当我看到这些工具时，我就想，要是早有这些，该多好。

But I'm looking at these tools, I'm like, man, that would have helped.

如果我重新回到应用数学领域，我一定会把这些工具用到极致。

And if I return to applied math, I'll be using the hell out of those tools.

以我现在的经验来看，这些工具能让我在成果的质量、全面性和产出速度上直接提升两倍。

I think it would make me just knowing what I know from doing this type of work, 2x more effective in terms of quality comprehensiveness and speed with which I produce results.

我觉得很多人没意识到，数学其实是创造性洞察力和大量繁琐工作的结合——学习并应用已有成果，尝试各种方法，推导细节，卡在某个地方，这在应用数学里太常见了：你已经有了想法，但卡在代数运算上。

I don't think people understand how much like mathematics is a combination of the creative insight plus a lot of tedious work, learning and applying results, trying different things, working out the details, getting stuck where you where you often get stuck in mathematics, at least in applied mathematics, is you have the idea, it's the algebra.

比如，我得到了某个随机变量的上界，

It's often like, I got a bound this like some of these random variables.

我需要把这个界控制在某个水平，才能用并界不等式，或者在这里用并界不等式，但变量之间有依赖关系，而且是相关的。

I need to bound it at this level to show that so I can apply a union bound and something like that, or I can apply a union bound here and there's dependencies, but they're they're correlated.

它们并不是完全负相关的，但我们可以用鞅来尽量保持这种非相关性。

They're not quite negative correlated, but we could apply like a martingale here to try to keep the discorrelation.

具体怎么操作，我有点记不清了。

And I don't quite remember how that works.

最顶尖的数学家，尤其是应用数学家，花更多时间去学习、练习和研读各种结论。

And the best mathematicians, applied mathematicians just spent more time learning results and practicing results and reading results.

如果能有AI帮你做这些事，那真的会带来巨大改变。

If you could have an AI help you with that, it really would make a big difference.

我是有切身体会才这么说的。

I'm telling you from experience.

如果AI能帮你探索证明空间，那将是极好的工具，未来必将成为数学研究的重要部分。

If you could have an AI explore proof spaces for you, it's a fantastic tool and it's gonna be a big part of mathematics.

我觉得不久的将来，会有一大批人去攻克那些容易出成果的问题。

And I think what's gonna happen is in the near future, you're gonna have an explosion of people doing like low hanging fruit results.

就像我们现在看到的这些论文里的情况。

Like we're seeing in these papers.

我正在自己专业领域内搜索子空间。

I'm searching subspaces within my specialty.

我只是在仔细调整AI，给它提出正确的问题。

I'm just like letting the AI I'm I'm, you know, very carefully tuning it and giving it the right questions.

比如，我们关注这个范围，但我会去发掘并完善这个领域里已有的很多成果。

Like, here's the space we're looking at, but I'm gonna kinda find, refine a lot of results that exist in the field.

这里有个小缺口。

There's a little gap here.

我们可以填补这个缺口。

We can close this gap.

我们发现了大量针对现有猜想的反例。

We're finding a lot of counter examples to existing like conjectures.

其中很多成果未必特别有意思，而且结果太多，审稿会成问题。

And a lot of it's not gonna be that interesting and there's gonna be a problem of too many results to be refereed.

但从中期来看，我觉得高端数学成果的整体平均质量会大幅提升。

But I think longer term, medium term, let's say, we're just gonna see the average quality of the high end math results is gonna jump up.

因为我确实之前出过一些挺有意思的成果。

Because I just know I produced a few interesting results in my time.

我觉得有了这些工具，它们还能更出色。

I think they could be more interesting with these tools.

所以我觉得数学的未来在这儿特别有前景，也合情合理。

So I think the future for mathematics here is exciting and it kinda makes sense.

数学和计算机编程本来就是一体的。

Math and computer program.

从一开始我们就被这么告诉过。

That's what we were told from the beginning.

大语言模型最擅长的就是这个。

LLMs sweet spot.

我们现在正开发各种定制工具和编程系统，比如复杂的代码框架，还有模块化设计的证明探索系统，专门用于数学研究。

And we're getting these custom tools, custom tools and programming, which are these like complicated coding harnesses and custom tools and mathematics, which are these modular architected sort of proof exploration systems.

这些工具已经产生了实实在在的影响。

And they're making a difference.