关于AI解决纳维尔-斯托克斯猜想

On AI Solving the Navier-Stokes Conjecture

2026年9月8日，OpenAI宣布其内部模型解决了数学界的千禧年问题之一：纳维尔—斯托克斯（NS）存在性与光滑性问题。具体而言，AI在光滑外力的情况下找到了有限时间内发生爆破的解，并且给出了Lean形式化验证。虽然最终的确认程序尚未完成，但这件事情已经基本不太可能被当作一次大模型幻觉。克雷数学研究所随后也对这一突破作出了积极回应。

On September 8, 2026, OpenAI announced that its internal model had solved one of mathematics’ Millennium Prize Problems: the Navier–Stokes (NS) existence and smoothness problem. Specifically, the AI found a solution that develops a finite-time blowup under smooth forcing, accompanied by formal verification in Lean. Although the final confirmation process remains incomplete, the result can now hardly be dismissed as another large-model hallucination. The Clay Mathematics Institute subsequently responded positively to the breakthrough. 

此事迅速引起了数学界的强烈反应。三天后，25位菲尔兹奖得主签署联名公开信，指出AI公司与数学共同体的目标存在严重错位。签署者包括陶哲轩、Peter Scholze、Cédric Villani等。Villani甚至以“世界末日”来形容这次冲击；Scholze则立刻加入了一个试图维护人类数学纯洁性的联盟。一个长期以来被数学界当作圣杯的问题得到了解答，随之而来的却是一场关于数学会不会失去意义的危机。这种反应本身，恐怕比NS问题的解答更能揭示当下数学共同体的某些深层问题。 

The announcement quickly provoked a strong reaction from the mathematical community. Three days later, 25 Fields Medalists signed an open letter declaring a severe misalignment between the goals of AI companies and those of the mathematical community. The signatories included Terence Tao, Peter Scholze, and Cédric Villani. Villani even described the upheaval as “the end of the world”; Scholze promptly joined an alliance seeking to preserve the purity of human mathematics. A problem long regarded as a holy grail of mathematics had been solved, yet what followed was a crisis over whether mathematics would lose its meaning. This reaction may reveal more about the mathematical community’s deeper problems than the solution to NS itself. 

数学家们给出了很多辩护。其中主要的一条是：数学并不只是解题，更重要的是理解、交流、提出重要问题，以及知识在一代代人之间的传承。孤零零地获得一个正确答案，并不等于数学真正取得了进步。 

Mathematicians have offered various arguments in their defense. One of the main ones is that mathematics is about more than solving problems: what matters more is understanding, communication, ask important questions, and the transmission of knowledge from one generation to the next. Obtaining a correct answer in isolation does not necessarily amount to genuine mathematical progress. 

这个说法当然成立。但如果沿着它继续追问，很快就会碰到一个颇为尴尬的问题：过去几十年，数学界究竟是按照什么标准来决定谁应该得到最高的认可？ 一个解决了著名猜想的人，和一个花了十年把艰深理论整理清楚、让更多人能够理解的人，谁更容易获得顶尖职位、重要奖项，以及整个圈子的尊崇？一个年轻人如果明确告诉导师，自己更愿意推动已有知识的理解与传承，而不是争取做出某个足以成名的突破，他会被认为找到了数学的本质，还是会被认为缺少做一流研究的潜力？这些问题的答案并不难猜。

This claim is certainly valid. But following it a little further quickly brings us to an awkward question: over the past several decades, what criteria has the mathematical community actually used to decide who deserves its highest recognition? Between someone who solves a famous conjecture and someone who spends ten years making a difficult theory clear and accessible, who is more likely to receive prestigious positions, major prizes, and the reverence of the entire field? If a young mathematician explicitly tells their advisor that they would rather advance the understanding and transmission of existing knowledge than pursue a breakthrough that could make their name, would they be seen as having grasped the essence of mathematics—or as lacking the potential to do first-rate research? The answers are not difficult to guess. 

至于提出重要问题这种说法，或许只有对数学圈、乃至整个学术圈不太了解的外部人员才会真的买账 -- 任何对这个圈子稍有深入了解的人都知道，除了极少数由于历史原因公认的圣杯级别的问题外，根本就没有什么问题是能够被公认为重要的。实际情况是，数学的几十上百个分支各自内部的“重要问题”，其他分支的大多数数学家根本就不感兴趣，甚至从来没听说过。而从另一方面来讲，如果“重要”指的是对领域的知识发展有很高的关键意义，而不涉及到从业者的主观感受因素，那么没有任何理由会认为AI无法识别并提出这样的问题 -- 这恐怕反而是AI最擅长的领域。

As for the claim that humans will still retain an advantage in posing important questions, this is probably something that only outsiders with limited familiarity with mathematics, or academia more broadly, would find convincing. Anyone with even a moderately deep understanding of how these fields actually operate knows that, apart from a tiny number of historically established “holy grail” problems, there is hardly any question whose importance is universally recognized. In reality, mathematics consists of dozens, if not hundreds, of subfields, each with its own set of important problems. Most mathematicians working in other areas are simply not interested in them, and in many cases have never even heard of them. From another perspective, if “important” means that a question has high objective significance for the development of knowledge in a field, rather than being important because researchers happen to find it subjectively interesting, then there is no obvious reason to believe that AI would be incapable of identifying and formulating such questions. If anything, this may be precisely the kind of task at which AI is especially well suited to excel.

因此，在AI突然开始攻克圣杯级问题的当口，数学界高强度地宣称“我们的目的从来不是解题”，这种表态就很难不让人怀疑。既然理解与传承才是目的，为什么这个目的长期没有得到相应的奖励？既然首证只是一种手段，为什么当这种手段被另一类主体掌握时，整个学科的意义突然就受到威胁了？ 

So when AI suddenly starts tackling holy-grail problems and the mathematical community loudly declares that “our purpose was never to solve problems,” it is difficult not to be suspicious. If understanding and transmission are the purpose, why have they gone so long without being rewarded accordingly? If being the first to prove something is merely a means to an end, why does the meaning of the entire discipline suddenly come under threat when another kind of agent masters that means? 

这暴露出的是数学界的自我描述与实际运转之间的距离。平时用于塑造英雄的标准，在新的竞争者占据优势之后，突然被解释成无足轻重的表面现象；那些长期处于次要地位的工作，则被迅速抬升为学科的真正灵魂。这样的转向可以被包装成对数学本质的重新认识，但如果没有相应的制度改变，它也完全可以被理解为一次维护既有地位的话语调整。 

What this exposes is the distance between the mathematical community’s account of itself and the way it actually operates. The standards routinely used to create heroes are suddenly recast as superficial once a new competitor gains the upper hand; work long relegated to a secondary position is swiftly elevated into the discipline’s true soul. This shift can be presented as a rediscovery of the essence of mathematics. But without corresponding institutional change, it can just as readily be understood as a rhetorical adjustment to protect established status. 

而另一条常见的辩护是：人类在攻克难题的过程中所经历的挣扎，以及由此产生的新方法、新概念和各种副产品，才是数学发展的主线。AI过快地把问题解决了，反而会让这些东西失去生长的机会。联名公开信所表达的担忧，很大程度上也沿着这个方向展开。 

Another common argument is that the struggles people endure while tackling difficult problems—and the new methods, concepts, and incidental discoveries that emerge along the way—are the main story of mathematical development. By solving problems too quickly, AI supposedly deprives these things of the opportunity to grow. Much of the concern expressed in the open letter follows this line of reasoning. 

这个论证在逻辑上同样有一个非常明显的缺口：AI给出了答案，怎么就妨碍人继续思考了？ AI证明了一个定理，人依然可以重新走一遍，可以寻找另一种证明，可以研究原有证明为何如此复杂，可以把其中零散的技巧整理成一种更普遍的方法。甚至，人还可以构建一个更高层次的理论，让原先那个需要艰苦攻克的问题，变成新框架下一个几乎显而易见的推论。这里面哪一件事情，会因为AI先交出了答案而变得不可能？ 

Yet this argument has an equally obvious logical gap: how does AI providing an answer stop people from continuing to think? Once AI proves a theorem, people can still work through it themselves, seek another proof, investigate why the existing proof is so complicated, or organize its scattered techniques into a more general method. They can even build a higher-level theory in which the original, painstakingly solved problem becomes an almost obvious consequence. Which of these activities becomes impossible because AI delivers an answer first? 

更何况，如果真正重视的是理解，那么一个新证明的出现，本来就应该提供更多可以理解的东西。过去只能猜测的对象，如今有了明确的性质；过去不知道能否走通的路径，如今至少有一条已经被打通。人可以选择沿着它走，也可以试着另辟蹊径。数学对象没有消失，人的好奇心也没有被机器拿走。 

Moreover, if understanding is what truly matters, a new proof should give us more to understand. An object whose properties we could previously only conjecture now has definite features; a path whose viability was unknown has now been traversed. People can follow it or try to find another route. The mathematical objects have not disappeared, and machines have not taken away human curiosity. 

那消失的是什么？

What, then, has disappeared? 

是抢先解决这个问题、从而青史留名的机会。 

The opportunity to solve the problem first and secure a place in history. 

把这一点说清楚，整个争论就容易理解了。一个问题还没有被解决时，围绕它展开探索，有机会兑换成突破性的成果、顶尖期刊的论文、重要的职位、一生的声望。一旦问题被解决，同样的探索即使依然能够带来深刻的理解，其职业回报也会迅速下降。于是在现有体系中，“已经有人解决”很容易被翻译成“继续做下去不值得”。

Once this is made explicit, the entire dispute becomes easier to understand. While a problem remains unsolved, exploring it offers the possibility of a breakthrough, a paper in a leading journal, a prestigious position, and a lifetime of recognition. Once it is solved, the same exploration may still yield profound understanding, but its professional rewards drop sharply. Under the existing system, “someone has already solved it” is therefore easily translated into “it is no longer worth pursuing.” 

但这是一个激励机制的问题。为什么会被说成是AI妨碍了人类理解数学？

But this is a problem with incentives. Why is it being described as AI obstructing human mathematical understanding? 

如果一个年轻人花几年时间，把AI给出的复杂证明转化成一个清晰、优美、能够被广泛掌握的理论，并因此获得稳定的研究条件，那么AI解题过快究竟会损害什么？他照样需要挣扎，照样可能走弯路，照样会在这个过程中发展出自己的方法与直觉。公开信里那些值得珍视的东西，一样都不必消失。之所以很多人觉得它们会消失，是因为大家默认了：失去首证的可能性之后，这些工作就不再值得被充分奖励。

Suppose a young researcher could spend several years transforming an intricate AI-generated proof into a clear, elegant, and widely accessible theory, and receive stable research support for doing so. What exactly would be harmed by AI solving the problem too quickly? The researcher would still struggle, still take wrong turns, and still develop methods and intuitions of their own. Nothing cherished in the open letter would have to disappear. The reason so many people expect these things to vanish is that they take it for granted that, once the possibility of being first is gone, such work no longer deserves to be adequately rewarded. 

这才是问题真正的出处。在这种文化中，首证权承担着一种近乎终极激励的功能。理解、交流与传承当然重要，但它们往往需要借助首创的光环，才能获得足够高的地位。AI所带来的冲击，把这种平时被高尚叙事遮蔽的结构暴露了出来。 

This is the real source of the problem. In this culture, priority of proof functions as something close to the ultimate incentive. Understanding, communication, and transmission certainly matter, but they often need the halo of originality to attain sufficiently high standing. The shock brought by AI has exposed a structure normally concealed beneath lofty narratives. 

从上面的逻辑出发，数学界当前更应该去做的，或许是重新审视自己制定的规则。谁应该得到职位，什么工作值得资助，什么样的数学人生应该被视为成功，这些标准有相当一部分就掌握在数学共同体自己手中。长期按照突破与首创来分配核心荣誉，却在机器开始取得突破之后，把理解与传承的困境全部归咎于外部技术，这在逻辑上是说不过去的。 事实上，已经有人开始正面讨论这种改变。Grant Sanderson、陶哲轩等人就提出应当提高解释性工作在招聘和终身教职评价中的地位。这恰好说明真正可以改变的东西一直存在。问题在于，这些想法究竟会落实到多少具体选择中？至于AI公司在成果归属、引用和研究公开方式上的行为，自然应该接受审查。但这些事情不能替数学界回答上述问题。即使明天所有公司都完美遵循学术规范，AI依然可能以越来越快的速度解决越来越多的问题。到那个时候，数学家们究竟是在保卫人的理解，还是在保卫只有少数人才有资格获得的英雄位置？

Following this logic, what the mathematical community should be doing now is reconsidering the rules it has established for itself. Who deserves a position, what work merits funding, and what constitutes a successful mathematical life—these standards are, to a considerable extent, in the community’s own hands. Having long distributed its highest honors according to breakthroughs and priority, it cannot reasonably blame external technology for all the difficulties facing understanding and transmission once machines begin producing breakthroughs. Some people have, in fact, started addressing this change directly. Grant Sanderson, Terence Tao, and others have proposed giving expository work greater weight in hiring and tenure decisions. This demonstrates that opportunities for change have been there all along. The question is how often these proposals will shape actual decisions? AI companies’ conduct regarding attribution, citation, and the disclosure of research should, of course, be scrutinized. But none of that answers these questions for the mathematical community. Even if every company followed academic norms perfectly tomorrow, AI could still solve more and more problems at an accelerating pace. At that point, are mathematicians defending human understanding, or defending the heroic standing available only to a select few? 

数学是一个高度被ego与天才神话支撑起来的学科。它的文化很容易让人相信，自己能够看见别人看不见的结构，能够跨越别人跨不过去的障碍，而这种稀缺的能力就是自身价值的根基。对于在这套叙事中成长、投入了几十年人生的人来说，机器突然开始占据原先属于顶尖天才的位置，当然会带来极为真实的痛苦。这种痛苦需要时间消化，但痛苦并不能自动使所有辩护都变得合理；尤其是当一个共同体开始把自身地位的动摇解释为整个知识世界的衰败时，尤其需要接受更严格的审视。NS问题只是一个具有如此强烈象征意义的节点；AI接下来大概率还会继续迅速变强。今天人们可以说，AI只是会解题，还不会建立理论；等到它能够建立理论，又可以说它还不会提出真正重要的问题。如此这般不断往后退，最后才发现，人生多少大好时光，浪费在一场无谓地保卫ego的注定失败的挣扎之中。

Mathematics is a discipline sustained to a remarkable degree by ego and the mythology of genius. Its culture readily encourages people to believe that they can perceive structures others cannot see, overcome obstacles others cannot cross, and that this rare ability is the foundation of their worth. For someone who has grown up within this narrative and invested decades of life in it, machines suddenly occupying territory once reserved for the greatest minds will naturally cause very real pain. That pain takes time to process. But pain does not automatically make every defense reasonable. A community deserves especially close scrutiny when it begins to portray the destabilization of its own standing as the decline of the entire world of knowledge. NS is merely one milestone with unusually powerful symbolic significance; AI will probably continue to improve rapidly. Today, people can say that AI merely solves problems and cannot build theories. Once it can build theories, they can say that it cannot ask truly important questions. Retreating again and again, they may eventually discover how much of their lives they have squandered on a futile, doomed struggle to defend their egos. 

只要这个前提没有改变，焦虑就不会结束。 

As long as this premise remains unchanged, the anxiety will never end. 

问题真正的核心并非AI最终会不会取代数学家，数学会不会消失。而是：在公开信中所描述的那种更强调主体性的社群文化，能够多快得到认可？多快落实到现实运转之中？数学界如果继续把主要精力放在向外追究谁破坏了原先的游戏，却不愿修改自己给参与者设定的奖励，那么真正损害数学未来发展的，恐怕会是这种对旧秩序的执着。甚至有不少对AI友好派的讨论中，也已经开始出现一种换汤不换药的倾向：AI让证明变得廉价，我们后面需要在评价体系中更强调理解/传承/写作；但却不去追问：为什么一定要保留那个把“评价”和“排序”当作是核心要素来组织共同体的文化？这个文化对于这个共同体真就那么重要吗？ 

The central question is not whether AI will ultimately replace mathematicians, or whether mathematics will disappear. It is how quickly the kind of community culture described in the open letter—one that places greater emphasis on human subjectivity—can gain acceptance and become part of how the field actually operates. If the mathematical community continues to devote most of its energy to identifying who disrupted the old game, while refusing to change the rewards it has established for its participants, then this attachment to the old order may be what truly damages the future development of mathematics. Even among discussions that are relatively friendly toward AI, there is already a tendency to reproduce the same basic structure under a different name: if AI makes proofs cheap, then perhaps future evaluation systems should place greater emphasis on understanding, transmission, exposition, or writing. But this still leaves a more fundamental question untouched: why must the community continue to be organized around a culture in which evaluation and ranking are treated as central organizing principles in the first place? Is that culture really so essential to the mathematical community?

当然，这也不只是数学面临的问题。但这种非唯一性对数学界而言无疑是个好事。

Of course, mathematics is not alone in facing this problem. For the mathematical community, this non-uniqueness is undoubtedly good news.