Mathematics has changed. Let's act like it

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Mathematics has changed. Let’s act like it

Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

关于我的研究和说明性论文的更新、开放问题的讨论以及其他与数学相关的主题。作者：陶哲轩

1 November, 2026 in guest blog, opinion | Tags: Frank Calegari | by Terence Tao

2026年11月1日，发布于客座博客，观点 | 标签：Frank Calegari | 作者：陶哲轩

[This is a guest post by Frank Calegari. This blog post was initially written in a different file format and converted using AI. — T.]

[这是一篇由 Frank Calegari 撰写的客座文章。这篇博文最初是用不同的文件格式编写的，并使用人工智能进行了转换。—— T.]

Why we should not be sad

为什么我们不应该感到悲伤

The release by OpenAI on October 6 of over 700 research papers was a watershed in the history of mathematics. AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics. So why are some mathematicians so unhappy?

OpenAI 在 10 月 6 日发布的 700 多篇研究论文是数学史上的一个分水岭。不可否认，人工智能现在有能力在数学的许多领域做出突破性的贡献。那么，为什么一些数学家会如此不开心呢？

Some of it is the apparent disregard by AI companies for the practice of mathematics. The OpenAI papers — despite some tremendous ideas — are atrociously written and violate our community expectations of attribution. Some of them are even wrong. We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.

部分原因在于人工智能公司对数学实践的明显漠视。OpenAI 的这些论文——尽管包含一些极好的想法——写得非常糟糕，且违背了我们社区对归属权的期望。其中一些甚至是错误的。相比于控制万亿级公司的行为，我们对如何聘用数学家、发表论文和培养学生拥有更大的掌控力，而这才是我们应该投入更多精力的地方。

Much of the reaction seems to reflect a feeling that the subject that they love is under attack, and is being “hollowed out”. The recent OpenAI dump made barely a dent on the general problems of functoriality and reciprocity in the Langlands program. But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry. Mathematicians should do a better job explaining that when we talk about the central problems in any area, we are merely discussing the most natural boundaries of our understanding — and that unlocking them is not the end but the first step to finding new playgrounds to explore. Of course, this takes time. I looked for areas of mathematics which were “harder hit”. It is hard not to be struck by Hugo Duminil-Copin’s lament:

许多反应似乎反映出一种感觉，即他们所热爱的学科正受到攻击，并正在被“掏空”。最近 OpenAI 的大规模发布对朗兰兹纲领中函子性和互反性的一般性问题几乎没有产生任何影响。但即使它解决了朗兰兹纲领中的每一个问题，认为这就“解决了数论”也和说元素周期表的构建解决了化学一样天真。数学家们应该更好地解释，当我们谈论任何领域的核心问题时，我们只是在讨论我们理解力中最自然的边界——而解开这些边界并不是终点，而是寻找新探索领域的起点。当然，这需要时间。我寻找了那些“受灾更严重”的数学领域。很难不被 Hugo Duminil-Copin 的哀叹所触动：

Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.

在我整个职业生涯中公开提到过的每一个重大开放性问题（无论是在演讲、讲座、文章还是资助申请中），都没有在这次公告中被遗漏。所有问题都被声称已经得到证明。

I expected to see a few of them in the list. But not all of them. Not all at once. Not with such nonchalance.

我预料到名单中会出现其中的几个。但不是全部。不是一次性全部出现。更不是以如此漫不经心的态度。

How would I respond to that?

我该如何回应呢？

The classification of finite simple groups.

有限单群的分类。

I can’t say much about percolation theory, but there is another historical precedent I think mathematicians should take note of. In 1972, Dan Gorenstein announced a program towards the classification of finite simple groups. After a flurry of activity over around a decade, the “classification” was announced by Gorenstein in 1983 (somewhat prematurely, with the case of quasithin groups not being completed for another two decades by Aschbacher and Smith).

对于渗流理论我无法多言，但有一个历史先例我认为数学家们应该注意。1972 年，Dan Gorenstein 宣布了一项关于有限单群分类的计划。经过大约十年的密集活动，Gorenstein 在 1983 年宣布了“分类”完成（这有些操之过急，因为拟薄群的情况直到二十年后才由 Aschbacher 和 Smith 完成）。

The classification of finite simple groups was in many ways a huge success, but it was also a failure in that it — apologies to my group theorist friends and collaborators — cooled a lot of the interest around the subject. The “famous problem” had gone, and mathematicians on the whole turned their attention elsewhere. And yet, there remain very rich questions in finite group theory and even about the classification itself (to name just one: John Conway’s favorite question is why does the Monster group exist?). Perhaps for some fields, the OpenAI dump feels like how a group theorist may have felt in 1970 if the classification was dumped in their office, not in the form of the actual papers, but printed out on a spirit duplicator, smudged a little, then spread randomly like confetti all over the floor.

有限单群的分类在许多方面是一个巨大的成功，但它也是一个失败，因为它——向我的群论朋友和合作者致歉——冷却了人们对该学科的许多兴趣。“著名问题”消失了，数学家们总体上将注意力转向了别处。然而，有限群论中甚至关于分类本身仍然存在非常丰富的问题（仅举一例：John Conway 最喜欢的问题是为什么魔群会存在？）。也许对于某些领域来说，OpenAI 的这次发布给人的感觉，就像 1970 年的群论学家如果发现分类结果被扔在办公室里一样——不是以正式论文的形式，而是用油印机印出来，弄得有些模糊，然后像五彩纸屑一样随机散落在地板上。

Right now, we need efforts like that of Gorenstein, Lyons, and Solomon and their work on the revised classification. If this was all we had to do, then it would be a fairly thankless task. But we also have the responsibility not to pronounce the subject “done” but rather — with these new tools available — ripe for further investigation. When mathematicians lament the winding paths missed by the construction of a galactic bypass, they need to realize that those old paths still exist and seek them out. And we need to reward that behavior when it comes to offering people jobs.

现在，我们需要像 Gorenstein、Lyons 和 Solomon 在修订分类方面所做的那样努力。如果这就是我们所要做的全部，那将是一项相当吃力不讨好的任务。但我们也有责任不要宣布该学科已经“完成”，而是——利用这些新工具——使其成熟以供进一步研究。当数学家们哀叹因建造银河旁路而错过的蜿蜒小径时，他们需要意识到那些古老的小径依然存在，并去寻找它们。当我们在提供工作机会时，我们需要奖励这种行为。

The promise of AI, and the (dis)-unity of mathematics:

人工智能的前景，以及数学的（不）统一：

Since I was 15, I’ve wanted to know what $\zeta(3)$ was. First, there was the childhood fantasy of finding a formula that Euler missed. But soon one learns more: how Grothendieck’s period conjecture, together with a sufficiently fine understanding of the geometry of mixed Tate motives (following work of Don Zagier, Sasha Goncharov, Francis Brown, and others) gives a completely satisfying structural, conjectural explanation of what’s going on. The irrationality of $\zeta(3)$, or even its transcendence, is a mere shadow of the actual interesting underlying problem. Apéry’s proof of the irrationality of $\zeta(3)$ has inspired me for over 30 years not because of its conclusion, but for how it reveals the tiniest hints of some still secret story relating algebraic $K$-functions to arithmetic.

从 15 岁起，我就一直想知道 $\zeta(3)$ 是什么。起初，这只是童年时期想要找到欧拉所遗漏公式的幻想。但很快人们就会学到更多：Grothendieck 的周期猜想，连同对混合 Tate 动机几何的足够精细的理解（继 Don Zagier、Sasha Goncharov、Francis Brown 等人的工作之后），如何对正在发生的事情给出了一个完全令人满意的结构性、猜想性的解释。$\zeta(3)$ 的无理性，甚至它的超越性，都只是实际有趣的潜在问题的一个影子。Apéry 对 $\zeta(3)$ 无理性的证明激励了我 30 多年，不是因为它的结论，而是因为它揭示了将代数 $K$-函数与算术联系起来的某些仍然隐秘的故事的微小线索。

I hope AI can help mathematicians break out of the straitjacket of being constrained by the first natural conjecture they cannot prove, and work on unraveling the deeper mysteries woven throughout our subject. In the last thirty years, progress in mathematics by humans has been remarkable. But at the same time, I have felt a growing unease; the number of times I have seen mathematicians of all levels being unable to explain to someone even slightly outside their field why what they are doing is interesting has been distressing. The natural grooves of our profession have very much been bent toward more narrow technical expertise than a bigger vision. AI allows us an amazing opportunity to broaden ourselves as mathematicians.

我希望人工智能能够帮助数学家们摆脱被第一个无法证明的自然猜想所束缚的紧身衣，并致力于解开编织在我们学科中的更深层次的奥秘。在过去的三十年里，人类在数学方面取得了显著的进步。但与此同时，我感到一种日益增长的不安；我看到各级数学家无法向稍微超出其领域的人解释为什么他们所做的事情是有趣的，这种情况令人苦恼。我们职业的自然轨迹已经更多地转向了狭窄的技术专长，而不是更宏大的愿景。人工智能为我们提供了一个绝佳的机会，让我们作为数学家能够拓宽自己的视野。

I have tried (not always successfully) to be broad as a mathematician. I had some help by the fact that, while I knew since I was 17 that I wanted to work in algebraic number theory and the Langlands progr

我一直试图（并不总是成功）成为一名视野广阔的数学家。我得到了一些帮助，因为虽然我从 17 岁起就知道我想从事代数数论和朗兰兹纲领的研究。