The AI Revolution in Math Has Arrived

T he tipping point came in the summer of 2025.

转折点出现在 2025 年夏天。

That July, several artificial intelligence models solved five out of six problems at the International Mathematical Olympiad, an annual challenge for some of the world’s best high school students.

那年 7 月，多个人工智能模型解决了国际数学奥林匹克竞赛中六个问题中的五个，这是世界上一些最优秀高中生的年度挑战。

But while mathematicians were shocked — few had expected the programs to get that good that quickly — the impressive results didn’t necessarily mean that AI would make important strides in research math.

然而，尽管数学家们感到震惊——很少有人预料到这些程序能这么快取得如此成果——令人印象深刻的结果并不一定意味着人工智能将在研究数学方面取得重要进展。

After all, Olympiad problems are challenging puzzles with known answers, not open questions.

毕竟，奥数题是具有已知答案的挑战性难题，而不是开放性问题。

Nevertheless, the results made people pay attention.

尽管如此，结果还是引起了人们的关注。

Mathematicians who had dismissed AI models as too error-prone to be useful started playing around with them.

那些认为人工智能模型太容易出错而无用的数学家们开始研究它们。

Those early adopters found, to their surprise, not only that the models were good at puzzles, but that they could help break genuinely new ground.

那些早期采用者惊讶地发现，这些模型不仅擅长解谜，而且可以帮助开辟真正的新天地。

Soon, mathematicians were using AI to discover and prove new results, accomplishing in a day what would have once taken them weeks or months.

很快，数学家们开始使用人工智能来发现和证明新结果，在一天内完成了以前需要数周或数月才能完成的工作。

“2025 was the year when AI really started being useful for many different tasks,” said Terence Tao , a prominent mathematician at the University of California, Los Angeles.

“2025 年是人工智能真正开始对许多不同任务发挥作用的一年，”加州大学洛杉矶分校著名数学家陶哲轩 (Terence Tao) 说。

AI is rapidly changing how mathematicians do math.

人工智能正在迅速改变数学家的数学计算方式。

They must now reckon with why they do it.

他们现在必须考虑为什么要这样做。

While no single new result is a world-beating breakthrough, some of them are on par with discoveries published in professional mathematical journals.

虽然没有一项新成果是世界级的突破，但其中一些成果与专业数学期刊上发表的发现相当。

In some cases, algorithms formulate a conjecture, prove it, and verify the proof with minimal human intervention.

在某些情况下，算法会提出一个猜想，证明它，并以最少的人为干预来验证证明。

In others, extensive chats with large language models such as ChatGPT, Claude, or Gemini lead to novel proof strategies.

在其他情况下，与 ChatGPT、Claude 或 Gemini 等大型语言模型的广泛聊天会产生新颖的证明策略。

“This guy’s got a shovel.

“这家伙有一把铲子。

This guy’s got a pickax.

这家伙有一把镐。

Together we can bore a tunnel,” Tao said.

我们可以一起挖隧道，”陶说。

There’s “a lot of throwing things at the wall to see what sticks.”

“很多人都会把东西扔到墙上，看看什么能粘住。”

Though Tao is perhaps the most prominent exponent of AI’s utility in mathematics, others agree.

尽管陶可能是人工智能在数学中的应用的最杰出的倡导者，但其他人也同意这一点。

Even by solving easy problems, said Daniel Litt of the University of Toronto, AI “is changing how mathematics is done.”

多伦多大学的丹尼尔·利特 (Daniel Litt) 表示，即使通过解决简单的问题，人工智能也“正在改变数学的计算方式”。

Soon, “it will look and feel altogether different from the way mathematics was traditionally done,” Tao said.

很快，“它看起来和感觉起来都将与传统的数学计算方式完全不同，”陶说。

Where before mathematicians studied one problem at a time, “with these tools you can solve thousands of problems at once and start doing statistical studies.” Though nobody I spoke with thinks AI will replace mathematicians, Tao added that “there are a lot of institutional changes, cultural changes, we will have to make.”

在数学家一次研究一个问题之前，“使用这些工具，您可以一次解决数千个问题并开始进行统计研究。”尽管与我交谈的人中没有人认为人工智能会取代数学家，但陶补充道，“我们必须做出很多制度变革和文化变革。”

Terence Tao is excited by the opportunities that AI models afford mathematicians.

陶哲轩对人工智能模型为数学家提供的机会感到兴奋。

Soon, math “will look and feel altogether different from the way mathematics was traditionally done,” he said.

他说，很快，数学“看起来和感觉起来都将与传统数学的方式完全不同”。

Those changes will be contested, in math as in other academic disciplines wrestling with AI’s impact.

这些变化将受到争议，无论是在数学领域，还是在其他学科领域，都在与人工智能的影响作斗争。

As AI models become a powerful new tool, they risk causing mathematicians to lose direct experience with mathematical understanding, said Akshay Venkatesh of the Institute for Advanced Study.

高等研究院的阿克谢·文卡特什 (Akshay Venkatesh) 表示，随着人工智能模型成为一种强大的新工具，它们可能会导致数学家失去数学理解的直接经验。

Like Tao, Venkatesh is a recipient of the Fields Medal, math’s most prestigious prize.

和陶一样，文卡特什也是数学界最负盛名的奖项菲尔兹奖的获得者。

Both agree that AI’s impact will be significant, but Venkatesh is more cautious about it: “There are valuable things in our culture which we should try to keep,” he said.

两人都认为人工智能的影响将是巨大的，但 Venkatesh 对此更加谨慎：“我们的文化中有一些有价值的东西，我们应该尽力保留，”他说。

Some mathematicians are now leaving academia to work at big tech firms, like OpenAI and Google, or to join math-focused AI startups such as Harmonic , Logical Intelligence , Axiom Math , and Math Inc. “One reason there is so much interest in AI for mathematics in the corporate world is that people are recognizing that the key to general intelligence is combining the insights you get from machine learning and the precision you get from mathematics,” said Jeremy Avigad , the director of the Institute for Computer-Aided Reasoning in Mathematics at Carnegie Mellon University.

一些数学家现在离开学术界，去 OpenAI 和 Google 等大型科技公司工作，或者加入 Harmonic、Logical Intelligence、Axiom Math 和 Math Inc 等专注于数学的人工智能初创公司。卡内基梅隆大学数学计算机辅助推理。

By the start of 2026, shock at the power of AI had turned into something more like wonder.

到 2026 年初，对人工智能力量的震惊已经变成了更多的奇迹。

A February challenge called First Proof gave entrants a week to have their AI models solve 10 research-level questions in various areas of math.

2 月份，一项名为“First Proof”的挑战赛为参赛者提供了一周的时间，让他们的人工智能模型解决各个数学领域的 10 个研究级问题。

Mathematicians had chosen the questions so that they were unlikely to have appeared in the algorithms’ training data.

数学家选择了这些问题，以便它们不太可能出现在算法的训练数据中。

With varying levels of autonomy, the models succeeded in solving over half the problems.

通过不同程度的自主权，这些模型成功解决了一半以上的问题。

If the Olympiad results represented the moment AI entered an ambitious college math program, the First Proof results were arguably the moment they finished graduate school.

如果说奥林匹克竞赛的结果代表了人工智能进入雄心勃勃的大学数学课程的那一刻，那么第一个证明的结果可以说是他们完成研究生院的那一刻。

In a blog post analyzing the results , Litt wrote: “It’s very likely that this technology is bigger than the computer .”

在分析结果的博客文章中，利特写道：“这项技术很可能比计算机更强大。”

Though the summer of 2025 marked an inflection point in the capabilities of AI, it didn’t come out of nowhere.

尽管 2025 年夏天标志着人工智能能力的拐点，但它并不是凭空出现的。

Pushmeet Kohli, Google DeepMind’s vice president of science, said DeepMind has been trying to solve math problems with AI since 2018.

谷歌 DeepMind 科学副总裁 Pushmeet Kohli 表示，自 2018 年以来，DeepMind 一直在尝试用人工智能解决数学问题。

François Charton , now at Axiom, first started trying to use machine learning to solve mathematical problems back in 2019 .

现供职于 Axiom 的 François Charton 于 2019 年首次尝试使用机器学习来解决数学问题。

But in those early years, it was a niche area.

但在早些年，这是一个小众领域。

At first, Charton and a handful of others used AI to solve problems whose solutions were already known, just to see if they could get the new techniques to work.

起初，查顿和其他一些人使用人工智能来解决已知解决方案的问题，只是为了看看他们是否能让新技术发挥作用。

By 2024, they were beginning to forge ahead.

到2024年，他们开始奋进。

They looked for problems where there was a rich set of data to analyze, and then used AI to construct mathematical objects with quantifiable properties — like optimal arrangements of points that could fit on a grid without forming an isosceles triangle .

他们寻找有丰富数据可供分析的问题，然后使用人工智能构建具有可量化属性的数学对象，例如可以适合网格而不形成等腰三角形的点的最佳排列。

Mark Belan/ Quanta Magazine; source: https://arxiv.org/abs/2411.00566

马克·贝兰/广达杂志；来源：https://arxiv.org/abs/2411.00566

In January 2025, Tao and Javier Gómez-Serrano of Brown University began working with two mathematicians at DeepMind, Adam Wagner and Bogdan Georgiev , on an AI system called AlphaEvolve.

2025 年 1 月，布朗大学的 Tai 和 Javier Gómez-Serrano 开始与 DeepMind 的两位数学家 Adam Wagner 和 Bogdan Georgiev 合作开发一个名为 AlphaEvolve 的人工智能系统。

AlphaEvolve works by using Gemini to write programs in Python code that might be hundreds of lines long.

AlphaEvolve 的工作原理是使用 Gemini 用 Python 代码编写可能长达数百行的程序。

It then “evolves” these programs using so-called genetic algorithms to attempt to find optimal solutions to math problems.

然后，它使用所谓的遗传算法“进化”这些程序，试图找到数学问题的最佳解决方案。

The four mathematicians used AlphaEvolve on a new problem every day or two for a few months.

四位数学家在几个月的时间里每天或每两天都使用 AlphaEvolve 来解决新问题。

As they did so, they also learned how to improve the prompts they gave AlphaEvolve.

在这样做的同时，他们还学会了如何改进给 AlphaEvolve 的提示。

One key takeaway: The model seemed to benefit from encouragement.

一个关键要点：该模型似乎受益于鼓励。

It worked better “when we were prompting with some positive reinforcement to the LLM,” Gómez-Serrano said.

戈麦斯-塞拉诺说，“当我们对法学硕士进行一些积极的强化时”效果更好。

“Like saying ‘You can do this’ — this seemed to help.

“就像说‘你能做到’——这似乎有帮助。

This is interesting.

这很有趣。

We don’t know why.”

我们不知道为什么。”

Javier Gómez-Serrano and his colleagues have used an AI system called AlphaEvolve to make progress on dozens of math problems.

Javier Gómez-Serrano 和他的同事使用名为 AlphaEvolve 的人工智能系统在数十个数学问题上取得了进展。

By late May, the team had tried AlphaEvolve on 67 different problems in several areas of mathematics.

到 5 月底，该团队已尝试 AlphaEvolve 解决多个数学领域的 67 个不同问题。

On 23 of them, AlphaEvolve improved, in a small way, on the best known solutions.

AlphaEvolve 对其中 23 个最知名的解决方案进行了小幅改进。

On 36 of the 67, it did as well as what had already been done, and on the remaining handful, it couldn’t match the best known result.

在 67 个中的 36 个中，它的表现与已经完成的一样好，而在剩下的少数几个中，它无法与最著名的结果相匹配。

The mathematicians shared their findings in a November 2025 paper, “ Mathematical Exploration and Discovery at Scale .” Gómez-Serrano noted that any one of their results might have been obtained by an expert in a given area who worked at it for a few months.

数学家们在 2025 年 11 月发表的论文《大规模数学探索与发现》中分享了他们的发现。戈麦斯-塞拉诺指出，他们的任何一项结果都可能是由某一特定领域的专家在该领域工作了几个月后获得的。

But without being experts in many of these fields, “we were able to obtain comparable results in the span of a day or two,” he said.

但他说，虽然我们不是这些领域的专家，但“我们能够在一两天内获得类似的结果”。

As Tao put it, current AI models are “very good at scouring big lists of problems for low-hanging fruit.

正如陶所说，当前的人工智能模型“非常擅长从大量问题中寻找容易实现的目标。

It’s tedious and thankless and not something humans want to do.” He cautioned that models are achieving “scattered successes among a big sea of unreported failures.” But the successes are notable.

这是乏味且吃力不讨好的事情，不是人类愿意做的事情。”他警告说，模型正在“在大量未报告的失​​败中取得零星的成功”。但成功是显着的。

Gómez-Serrano estimates that he now spends about two-thirds of his time using AI.

戈麦斯-塞拉诺估计，他现在大约三分之二的时间都在使用人工智能。

It is, he said, “getting to the point where it is useful and usable.

他说，“它已经达到了有用和可用的地步。

This is the beginning of the new way we will do mathematics.”

这是我们数学研究新方法的开始。”

In previous years, AI’s extra power seemed to come from its ability to resurface long-forgotten proofs buried in obscure references.

前几年，人工智能的额外力量似乎来自于它能够重新呈现隐藏在模糊参考文献中长期被遗忘的证据。

Igor Pak of UCLA noted that ChatGPT is currently “fantastic in finding the right references, right literature, finding connections that Google Scholar — which doesn’t work semantically — can’t.”

加州大学洛杉矶分校 (UCLA) 的伊戈尔·帕克 (Igor Pak) 指出，ChatGPT 目前“在找到正确的参考文献、正确的文献、找到 Google Scholar（在语义上不起作用）无法找到的联系方面表现出色。”

Then, over the course of 2025, said Johannes Schmitt of the Swiss Federal Institute of Technology Zurich, something shifted.

瑞士苏黎世联邦理工学院的约翰内斯·施密特 (Johannes Schmitt) 表示，到了 2025 年，事情发生了变化。

“It started becoming useful to talk to LLMs, not because they would give you the full answer,” he said, but because “they became good conversation partners.”

“与法学硕士交谈开始变得有用，不是因为他们会给你完整的答案，”他说，而是因为“他们成为了很好的对话伙伴。”

Johannes Schmitt has recently noticed a rapidly growing role for AI in mathematics: as a conversation partner.

约翰内斯·施密特 (Johannes Schmitt) 最近注意到人工智能在数学中的作用迅速增强：作为对话伙伴。

The LLMs he spoke with inevitably made lots of mistakes, leading some mathematicians to dismiss them outright.

与他交谈过的法学硕士不可避免地犯了很多错误，导致一些数学家彻底驳回了他们。

Many researchers, he said, decide that if “everything it says is kind of wrong, I will just not talk to it.” But others — he puts himself in this camp — have a higher tolerance for “the pain of talking to this bullshitting model.

他说，许多研究人员认为，如果“它所说的一切都是错误的，我就不会和它说话。”但其他人——他把自己归入这个阵营——对“与这个胡言乱语的模特交谈的痛苦”有更高的容忍度。

They say, I can still get something out of this conversation; even if not every idea is good, I can ignore the bad ones and take the good ones.” And the mistakes, Schmitt noted, are weird ones: There is virtually no way that a person with any training in mathematics would make such a plethora of basic errors while also succeeding in coming up with subtle, original, and correct ideas.

他们说，我仍然可以从这次谈话中得到一些东西；即使不是每个想法都是好的，我也可以忽略坏的想法并接受好的想法。”施密特指出，这些错误都很奇怪：一个受过数学训练的人几乎不可能在犯如此多的基本错误的同时，还能成功地提出微妙的、原创的和正确的想法。

UCLA’s Ernest Ryu , who works largely in a branch of applied math called optimization theory, also started paying more attention to LLMs after the Olympiad results.

加州大学洛杉矶分校 (UCLA) 的欧内斯特·柳 (Ernest Ryu) 主要研究应用数学的一个分支——优化理论，在奥赛成绩公布后，他也开始更加关注法学硕士。

Whereas AlphaEvolve was trying to optimize particular quantities, Ryu wanted to prove things about the conditions under which optimization algorithms work.

AlphaEvolve 试图优化特定数量，而 Ryu 想要证明优化算法工作的条件。

In the summer of 2025, he noticed that LLMs’ mathematical capabilities had dramatically improved.

2025年夏天，他注意到法学硕士的数学能力有了显着提高。

He started using them to help prepare lecture notes, mostly to fill in gaps in his memory of the details of a particular proof.

他开始使用它们来帮助准备讲义，主要是为了填补他对特定证明细节的记忆空白。

At times, he said, “it would find an error in my reasoning, sometimes major, sometimes minor.

他说，有时“它会发现我的推理中存在错误，有时是大错误，有时是小错误。

Sometimes it would find a simpler proof than I had in my notes.”

有时它会找到比我笔记中更简单的证明。”

Ernest Ryu recently parlayed conversations with ChatGPT into a proof of a decades-old conjecture.

Ernest Ryu 最近将与 ChatGPT 的对话变成了几十年前猜想的证明。

“The use of ChatGPT really accelerated the discovery,” he said.

“ChatGPT 的使用确实加速了这一发现，”他说。

He had a sense that AI models were “exhibiting signs of life.” He remembers feeling skeptical but optimistic.

他有一种感觉，人工智能模型正在“展现出生命的迹象”。他记得当时感到怀疑但乐观。

To make up his own mind about what LLMs can and can’t do, he decided to try an experiment.

为了确定法学硕士能做什么、不能做什么，他决定尝试一个实验。

One evening in October, after his young son went to sleep, he set out to solve an open problem in optimization theory that he’d attempted a few times in the past.

十月的一个晚上，他的小儿子入睡后，他开始解决优化理论中的一个开放性问题，他过去曾尝试过该问题几次。

This time he used ChatGPT.

这次他使用了ChatGPT。

“It’s not the most important problem, but I know 10 people who would very much appreciate a solution,” he said.

“这不是最重要的问题，但我知道有 10 个人非常希望找到解决方案，”他说。

Ryu’s problem was first proposed in 1983 by a Russian mathematician named Yurii Nesterov.

Ryu 问题最早于 1983 年由俄罗斯数学家 Yurii Nesterov 提出。

Nesterov was trying to find the minimum of functions that take many variables as inputs and output a single value that behaves “nicely” in a particular mathematical way.

Nesterov 试图找到将多个变量作为输入并输出以特定数学方式表现“良好”的单个值的函数的最小值。

If you think of the outputs as forming an elevation map, you want to prove that you eventually converge on the lowest point, and don’t end up endlessly bouncing around in search of it.

如果您将输出视为形成高程图，那么您想要证明您最终会收敛到最低点，并且不会最终无休止地跳来跳去寻找它。

This sort of problem arises quite often in applied math, especially in machine learning, where it is central to training neural networks.

这类问题在应用数学中经常出现，特别是在机器学习中，它是训练神经网络的核心。

Say you start somewhere on your map.

假设您从地图上的某个地方开始。

A widely used technique known as gradient descent uses basic tools from calculus to figure out which way is downhill and how steep the hill is at the point where you’re standing.

一种广泛使用的技术称为梯度下降，它使用微积分的基本工具来确定哪条路是下坡以及您所站位置的山坡有多陡。

Take a step downward in the steepest direction each time, and you will eventually get to the very bottom.

每次沿着最陡的方向向下走一步，最终就会到达底部。

But although gradient descent will get you to the right answer, it sometimes gets you there very slowly.

但是，虽然梯度下降会让你得到正确的答案，但有时它会让你非常缓慢地到达那里。

So mathematicians have long looked for variations that converge to the right answer more quickly.

因此，数学家长期以来一直在寻找能够更快地收敛到正确答案的变体。

Nesterov developed one technique for doing so, in which the size of each step downhill depends not only on how steep the function is at a given point, but also on the path you’ve already taken to get there.

Nesterov 为此开发了一种技术，其中每一步下坡的大小不仅取决于函数在给定点的陡峭程度，还取决于您到达那里已经采取的路径。

If you’ve been taking bigger steps in the past, you’ll continue to do so.

如果您过去已经采取了更大的步骤，那么您将继续这样做。

Yurii Nesterov’s conjecture about an optimization algorithm he’d developed stood open for decades.

尤里·内斯特罗夫（Yurii Nesterov）关于他开发的优化算法的猜想几十年来一直没有被公开。

It was only with the help of AI that it was resolved.

只有在人工智能的帮助下，这个问题才得以解决。

It seems intuitively obvious that this will get you to the bottom of the hill more quickly.

从直觉上看，这显然会让你更快地到达山脚。

But what if you go too fast and overshoot?

但是如果你走得太快并且超调了怎么办？

You might risk endlessly oscillating around the true minimum, and never attaining it.

你可能会冒着在真正的最小值附近无休止地摇摆的风险，并且永远无法达到它。

Nesterov couldn’t prove that his algorithm would eventually converge to the optimal value.

Nesterov 无法证明他的算法最终会收敛到最优值。

And for 42 years, no one else could either.

42 年来，没有人能做到这一点。

When Ryu asked ChatGPT, “it kept giving me incorrect proofs,” he said.

当 Ryu 询问 ChatGPT 时，“它一直给我提供错误的证明，”他说。

“But the lead-up to the inevitable error had interesting steps, correct partial results that seemed potentially useful.” As the LLM made incremental progress, he would check its answers, keep the correct parts, and feed them back into the model with a new prompt.

“但导致不可避免的错误的步骤很有趣，纠正了部分结果，似乎可能有用。”随着法学硕士取得渐进的进展，他会检查答案，保留正确的部分，并用新的提示将它们反馈到模型中。

“I had to play the role of the verifier,” Ryu said.

“我必须扮演验证者的角色，”Ryu 说。

“With ChatGPT, I felt like I was covering a lot of ground very rapidly, much more quickly than I could do on my own.

“通过 ChatGPT，我感觉自己很快就了解了很多领域，比我自己的速度快得多。

That’s what kept me going.”

这就是让我继续前进的动力。”

Within about 12 hours of work spread over three days, he had arrived at a proof of a simplified version of the problem.

经过三天大约 12 个小时的工作，他得出了该问题的简化版本的证明。

After a few more days, he finally proved that Nesterov’s method converges .

又过了几天，他终于证明了涅斯特罗夫的方法是收敛的。

It wasn’t, Ryu said, “the most creative thing, not the most complicated.

Ryu 说，“这不是最有创意的事情，也不是最复杂的事情。

But certainly it wasn’t that easy.” While it’s not a life-changing result, he added, “it’s something that could be published in a top optimization journal without the AI component.

但肯定没那么容易。”他补充道，虽然这不是一个改变生活的结果，但“这是可以在没有人工智能组件的情况下发表在顶级优化期刊上的结果。

It’s a good result.”

这是一个很好的结果。”

“It’s a concrete instance where the use of ChatGPT really accelerated the discovery,” he said.

“这是一个具体的例子，ChatGPT 的使用确实加速了这一发现，”他说。

And he thinks the capabilities of LLMs are only going to continue to improve.

他认为法学硕士的能力只会继续提高。

“ If you look at the velocity of the improvement, that’s staggering.

“如果你看看改进的速度，那是惊人的。

If we continue one year later, two or three model releases down the line, we are going to get really, really impressive, substantial discoveries assisted by AI.

如果我们继续一年后，发布两到三个模型，我们将在人工智能的帮助下获得非常非常令人印象深刻的实质性发现。

It’s going to come.”

它会来的。”

A few months after sharing his paper on Nesterov’s method, Ryu took a leave of absence from UCLA to take a job at OpenAI, where he is now a member of the technical staff.

在分享有关 Nesterov 方法的论文几个月后，Ryu 从加州大学洛杉矶分校请假前往 OpenAI 工作，现在是该公司的技术人员。

Over the course of 2025 and into the first months of 2026, AI has been used to prove increasingly abstract results.

从 2025 年到 2026 年前几个月，人工智能已被用来证明越来越抽象的结果。

In September 2025, more than 100 mathematicians from around the world gathered at Brown University for a special program on algebraic combinatorics .

2025年9月，来自世界各地的100多名数学家齐聚布朗大学，参加代数组合学的特别课程。

Nicolás Libedinsky and David Plaza had come from Chile, José Simental from Mexico, Geordie Williamson from Australia, and Jordan Ellenberg from Wisconsin.

尼古拉斯·利贝丁斯基和大卫·普拉扎来自智利，何塞·西门塔尔来自墨西哥，乔迪·威廉姆森来自澳大利亚，乔丹·埃伦伯格来自威斯康星州。

All of them were interested, for different reasons, in computing a quantity called the d -invariant, which appears in many areas of math.

出于不同的原因，他们所有人都对计算一个称为 d 不变量的量感兴趣，该量出现在数学的许多领域。

To understand what the d -invariant is, it helps to first look at a well-studied object in one of these areas, called the permutation group.

要理解 d 不变量是什么，首先查看这些领域之一中经过深入研究的对象（称为排列群）会有所帮助。

This object is a way of describing the different ways one can shuffle a set of items, like cards in a deck.

该对象是一种描述洗牌一组物品（例如一副牌中的牌）的不同方式的方式。

It starts out simple.

开始很简单。

If you have a deck with just one card, you can’t shuffle it.

如果你的一副牌只有一张牌，你就不能洗牌。

So the permutation group S 1 has one element.

因此置换群S 1 有一个元素。

S 2 has two elements: If you have two cards, they can appear in two possible orders.

S 2 有两个元素：如果您有两张牌，它们可以以两种可能的顺序出现。

S 3 becomes a little more complicated; there are six different ways to order a deck of three cards.

S 3 变得稍微复杂一些；有六种不同的方式来订购一副三张牌。

The different ways of ordering the cards can be arranged into a network of vertices and edges called a graph.

可以将卡片排序的不同方式排列成称为图的顶点和边的网络。

The starting arrangement, 123, goes at the bottom.

起始排列 123 位于底部。

Each edge of the graph (drawn as an arrow) represents a swap of two cards:

图表的每条边（绘制为箭头）代表两张牌的交换：

As the number of cards n gets larger, S n grows very quickly — making this graph next to impossible to draw for groups after S 4 .

随着牌数 n 变大，S n 增长得非常快——使得这张图几乎不可能为 S 4 之后的组绘制。

( S 60 has about as many elements as there are atoms in the observable universe.)

（S 60 的元素数量大约与可观测宇宙中的原子数量一样多。）

Mathematicians want to understand the structure of these graphs both as objects in and of themselves and as tools for analyzing other things.

数学家希望理解这些图的结构，既将其作为对象本身，又将其作为分析其他事物的工具。

Consider again the graph of the permutation group S 3 , which has six elements, or permutations.

再次考虑置换群 S 3 的图，它具有六个元素或置换。

We want to explore the relationship between these permutations.

我们想要探索这些排列之间的关系。

One way to do so is to look at all the ways to get from one permutation to another by following the arrows.

一种方法是按照箭头查看从一种排列到另一种排列的所有方法。

A given permutation is “smaller” than another one (using a definition of size called the Bruhat order) if it is possible to travel along the arrows from the first permutation to the second.

如果可以沿着箭头从第一个排列移动到第二个排列，则给定的排列比另一个排列“小”（使用称为 Bruhat 顺序的大小定义）。

So 213 is smaller than 321.

所以 213 比 321 小。

We can then look at the “Bruhat interval” between the two permutations — the set of all the different permutations that lie between them when you follow the graph’s arrows.

然后，我们可以查看两个排列之间的“Bruhat 区间”——当您沿着图表的箭头前进时，它们之间的所有不同排列的集合。

For example, the interval between 213 and 321 (seen below in red) includes 231 and 312.

例如，213 和 321 之间的区间（如下图红色所示）包括 231 和 312。

(If you can’t get from one permutation to another by following the arrows, like from 213 to 132, then neither one is smaller than the other, and the interval between them is not defined.)

（如果您无法按照箭头从一种排列到另一种排列，例如从 213 到 132，那么两个排列都不小于另一个排列，并且它们之间的间隔未定义。）

The d -invariant associated with two permutations is, loosely speaking, a measure of the complexity of their Bruhat interval’s underlying structure.

宽泛地说，与两个排列相关的 d 不变量是对其 Bruhat 区间基础结构复杂性的度量。

The same quantity appears in a number of mathematical questions that otherwise seem unrelated, making it of great interest to mathematicians.

同样的量出现在许多看似无关的数学问题中，这使得数学家们非常感兴趣。

In bigger permutation groups, it’s hard to say in any general way what the Bruhat interval between two given permutations looks like.

在较大的排列群中，很难以任何一般方式说明两个给定排列之间的 Bruhat 间隔是什么样的。

“Intervals are just super complicated things,” Libedinsky said.

“间隔是非​​常复杂的事情，”利贝丁斯基说。

He, Simental, Plaza, Williamson, and Ellenberg — all hoping for different reasons to find the biggest possible d -invariant for a given permutation group — set out to get AI to help them.

他、西门塔尔、普拉扎、威廉姆森和埃伦伯格——出于不同的原因都希望为给定的排列群找到最大可能的 d 不变量——开始让人工智能来帮助他们。

S ome mathematicians are concerned that AI will exacerbate existing inequalities.

一些数学家担心人工智能会加剧现有的不平等。

“It is not lost on me that the reason I had access to AlphaEvolve is because I did a semester in the U.S.,” said José Simental of the National Autonomous University of Mexico in Mexico City.

“我并没有忘记，我能够使用 AlphaEvolve 的原因是因为我在美国学习了一个学期，”位于墨西哥城的墨西哥国立自治大学的 José Simental 说道。

“I think we as a community have to be very careful on ensuring equitable access to these sort of tools.”

“我认为我们作为一个社区必须非常小心，确保公平地获得此类工具。”

They ended up finding something else entirely.

他们最终找到了完全不同的东西。

In October 2025, Ellenberg asked Wagner at DeepMind to use AlphaEvolve (which is not publicly available) to analyze the structures of the Bruhat intervals of dozens of permutation groups.

2025 年 10 月，Ellenberg 要求 DeepMind 的 Wagner 使用 AlphaEvolve（未公开）来分析数十个排列群的 Bruhat 区间的结构。

It ran overnight.

它运行了一夜。

“In the morning, we were like, this program is really doing something interesting,” Williamson said.

“早上，我们就想，这个项目确实在做一些有趣的事情，”威廉姆森说。

“And then I remember a flurry of emails back and forth that day.”

“然后我记得那天来来回回的一系列电子邮件。”

The LLM talked to itself while performing calculations.

LLM 在执行计算时自言自语。

“I’m about to propose something truly outlandish, a ‘Crazy Ivan’ maneuver for this problem,” it mused, referring to a sharp turn submarines sometimes take to detect their adversaries, popularized in the Tom Clancy novel The Hunt for Red October .

它若有所思地说：“我将提出一些真正古怪的办法，即针对这个问题的‘疯狂伊万’策略。”它指的是潜艇有时会采取急转弯来发现对手，这种策略在汤姆·克兰西的小说《追捕红色十月》中广为人知。

Ultimately, AlphaEvolve generated something like 50 lines of Python code in its attempts to find intervals with large d -invariants.

最终，AlphaEvolve 生成​​了大约 50 行 Python 代码，试图找到具有大 d 不变量的区间。

As the mathematicians tried to figure out what this code was doing, Ellenberg realized that if the number of cards in the deck was a power of 2 (like 16, which is 2 4 ), then the program became much shorter — about five lines long.

当数学家试图弄清楚这段代码的作用时，埃伦伯格意识到，如果这副牌中的牌数是 2 的幂（比如 16，即 2 4 ），那么程序就会变得更短——大约五行长。

“You can analyze it very explicitly,” Williamson said.

“你可以非常明确地分析它，”威廉姆森说。

“It’s doing something very beautiful.”

“它正在做一些非常美丽的事情。”

Clockwise from top left: Jordan Ellenberg, José Simental, Geordie Williamson, Nicolás Libedinsky, and David Plaza uncovered a surprising new structure in a well-studied mathematical object thanks to AI.

从左上角顺时针方向：Jordan Ellenberg、José Simental、Geordie Williamson、Nicolás Libedinsky 和 ​​David Plaza 通过人工智能在经过充分研究的数学对象中发现了令人惊讶的新结构。

Clockwise from top left: Geordie Williamson, Jordan Ellenberg, José Simental, Nicolás Libedinsky, and David Plaza uncovered a surprising new structure in a well-studied mathematical object thanks to AI.

从左上角顺时针方向：Geordie Williamson、Jordan Ellenberg、José Simental、Nicolás Libedinsky 和 ​​David Plaza 通过人工智能在经过充分研究的数学对象中发现了令人惊讶的新结构。

As they related in a preprint on January 3, 2026, AlphaEvolve had found that the Bruhat intervals in these particular permutation groups had a surprisingly special structure.

正如他们在 2026 年 1 月 3 日的预印本中所述，AlphaEvolve 发现这些特定排列群中的 Bruhat 区间具有令人惊讶的特殊结构。

When the researchers studied the intervals, they found that they formed higher-dimensional cubes called hypercubes.

当研究人员研究这些间隔时，他们发现它们形成了称为超立方体的更高维立方体。

“If you look at what AlphaEvolve was thinking, I was super surprised,” Libedinsky said.

“如果你看看 AlphaEvolve 的想法，我会感到非常惊讶，”Libedinsky 说。

“If it was a human, it would be an extremely creative human.”

“如果它是一个人，那将是一个极具创造力的人。”

AlphaEvolve had answered a question they didn’t know they had.

AlphaEvolve 回答了一个他们不知道的问题。

“We didn’t ask AlphaEvolve to find big hypercubes,” Ellenberg said.

“我们没有要求 AlphaEvolve 寻找大的超立方体，”埃伦伯格说。

“We asked it to find something else, and we thought about it and realized it was a gigantic hypercube which we had not anticipated was there.”

“我们要求它寻找其他东西，我们思考了一下，意识到它是一个巨大的超立方体，我们没有预料到它会在那里。”

Excerpt from a prompt that mathematicians gave to AlphaEvolve, in which they asked it to construct an object called a Kakeya set.

摘自数学家给 AlphaEvolve 的提示，他们要求 AlphaEvolve 构造一个称为 Kakeya 集的对象。

Mathematicians have found that AI performs better with encouragement.

数学家发现，人工智能在鼓励下表现得更好。

As Williamson put it, “It’s a structure that’s been sitting there for 50 years in front of our nose.

正如威廉姆森所说：“这个结构在我们眼前已经存在了 50 年。

We just hadn’t noticed it.”

我们只是没有注意到而已。”

Older machine learning methods had previously enabled such serendipitous mathematical discoveries, too — uncovering patterns no one had thought to look for.

较旧的机器学习方法以前也曾实现过这种偶然的数学发现——揭示了没有人想到寻找的模式。

But in the past, Williamson said, it was a “real engineering effort.

但威廉姆森说，在过去，这是“真正的工程努力”。

…

……

You need to know how to code, spend a lot of time looking at details of neural network training.

你需要知道如何编码，花大量时间查看神经网络训练的细节。

It was basically extremely difficult for a mathematician with no significant machine learning background to do this.”

对于没有重要机器学习背景的数学家来说，要做到这一点基本上是极其困难的。”

With LLMs, “I can suddenly do an experiment in 20 minutes that two years ago would have taken me two weeks,” he said.

有了法学硕士，“我突然可以在 20 分钟内完成一项实验，而两年前需要两周时间，”他说。

Though “most of the time it doesn’t work,” AI can now be used like never before “to discover the world that has riches beyond our imagination.”

尽管“大多数时候它不起作用”，但人工智能现在可以以前所未有的方式“发现拥有超乎我们想象的财富的世界”。

Though Bruhat intervals seem like purely combinatorial objects, they also play an important role in a particularly abstract area of math called algebraic geometry, which Ravi Vakil , a mathematician at Stanford University and the current president of the American Mathematical Society, specializes in.

尽管 Bruhat 区间看起来像是纯粹的组合对象，但它们在称为代数几何的特别抽象的数学领域中也发挥着重要作用，斯坦福大学数学家、现任美国数学会主席 Ravi Vakil 专门研究该领域。

Algebraic geometry is the study of shapes defined by polynomial equations like x 3 + 2 x 2 y + xz = 5, which involve a sum of variables raised to whole-number exponents.

代数几何是对由 x 3 + 2 x 2 y + xz = 5 等多项式方程定义的形状的研究，其中涉及到整数指数的变量之和。

The degree of the equation is the highest exponent the polynomial has, in this case 3.

方程的次数是多项式的最高指数，在本例中为 3。

Ravi Vakil and his colleagues recently came up with a novel proof idea while chatting with a bespoke version of Gemini.

Ravi Vakil 和他的同事最近在与定制版本的 Gemini 聊天时提出了一个新颖的证明想法。

“Who is that idea due to?” he asked.

“这个想法是谁提出来的？”他问道。

“Is it due to us?

“是我们的原因吗？

Is it due to the model?”

是因为型号的原因吗？”

Vakil and his colleagues, Balázs Elek of the University of New South Wales and Jim Bryan of the University of British Columbia, were interested in studying how spheres can be embedded in special spaces called flag varieties.

瓦基尔和他的同事——新南威尔士大学的巴拉兹·埃莱克（Balázs Elek）和不列颠哥伦比亚大学的吉姆·布莱恩（Jim Bryan）对研究如何将球体嵌入到称为“旗帜品种”的特殊空间中感兴趣。

(Flag varieties appear in the Bruhat team’s paper as well.) Each embedding — a way of associating each point on the sphere to a point within the flag variety — can be defined by a polynomial equation.

（旗帜变体也出现在 Bruhat 团队的论文中。）每个嵌入（一种将球体上的每个点与旗帜变体中的点相关联的方法）都可以通过多项式方程来定义。

There are lots of ways to embed the sphere.

嵌入球体的方法有很多种。

Mathematicians represent each embedding as its own point in a separate high-dimensional space.

数学家将每个嵌入表示为单独的高维空间中自己的点。

They then study the embeddings defined by polynomials of different degrees by analyzing the different spaces they form.

然后，他们通过分析它们形成的不同空间来研究由不同次数的多项式定义的嵌入。

As the degree increases, mathematicians want to understand how these spaces change.

随着程度的增加，数学家想要了解这些空间如何变化。

They knew that when the degree gets arbitrarily large — as it goes to infinity — the space resembles the space of all continuous embeddings, not just those defined by polynomials.

他们知道，当次数变得任意大时（当它达到无穷大时），该空间类似于所有连续嵌入的空间，而不仅仅是多项式定义的空间。

But when does this resemblance come to pass?

但这种相似之处是什么时候出现的呢？

AlphaEvolve talked to itself as it attempted to solve a problem about the Bruhat intervals of permutation groups.

AlphaEvolve 在试图解决有关排列群的 Bruhat 区间的问题时自言自语。

Vakil and his colleagues had found examples that suggested, to their surprise, that it happens very quickly.

瓦基尔和他的同事发现了一些例子，令他们惊讶的是，这种情况发生得非常快。

“There was some consistency that was not supposed to happen until you reached infinity, and it already happened,” he said.

“在达到无穷大之前，有些一致性是不应该发生的，但它已经发生了，”他说。

So, together with Freddie Manners and George Salafatinos , who were then working for DeepMind, they set out to prove it using two specialized modules built atop Google Gemini: DeepThink, which is publicly available, and a system developed by Salafatinos, called FullProof, which is not.

因此，他们与当时在 DeepMind 工作的 Freddie Manners 和 George Salafatinos 一起，开始使用在 Google Gemini 上构建的两个专用模块来证明这一点：DeepThink（公开可用）和 Salafatinos 开发的名为 FullProof（非公开）的系统。

They started with a simpler case.

他们从一个更简单的案例开始。

“The proof it gave was very elegant, correct, beautifully written.

“它给出的证明非常优雅、正确，写得很漂亮。

We could follow it line by line,” Vakil said.

我们可以逐行跟踪它，”瓦基尔说。

“It made clear a structure that was not obvious at the time.

“它明确了当时并不明显的结构。

From that, we realized how the whole argument and significant generalization should potentially work.”

由此，我们意识到整个论点和重要的概括应该如何发挥作用。”

Vakil and his colleagues then went back to the AI model, sketching a proof of the general case and asking it to fill in the details.

然后，瓦基尔和他的同事们回到人工智能模型，勾勒出一般情况的证明，并要求它填写细节。

As they reported in a preprint on January 12, 2026 , it succeeded.

正如他们在 2026 年 1 月 12 日的预印本中报道的那样，它成功了。

“To me,” Vakil said, “the real thing was the first thing” — DeepMind’s proof of the simpler case.

“对我来说，”Vakil 说，“真正的事情才是第一件事”——DeepMind 对更简单案例的证明。

“The clarity of the argument gave us a new idea.” But he wonders: “Who is that idea due to?

“论证的清晰性给了我们一个新的想法。”但他想知道：“这个想法是谁提出的？

Is it due to us?

是我们的原因吗？

Is it due to the model?”

是因为型号的原因吗？”

However one ascribes credit, Vakil said, “I believe I would have come up with the proof given enough time.”

不管如何归因，瓦基尔说：“我相信，如果有足够的时间，我会提出证明。”

But then he hesitated.

但随后他犹豫了。

“I think so.

“我想是的。

I’m not sure.

我不确定。

I don’t know.

我不知道。

Maybe I would have done it in a clunky way.

也许我会用一种笨拙的方式来做这件事。

Very possibly, the paper wouldn’t have happened without the assistance.”

如果没有这些帮助，这篇论文很可能就不会发表。”

And finally: “We needed to go back and forth.

最后：“我们需要来回走动。

AI models will help us do mathematics by letting us do things we did not have time to do before.”

人工智能模型将帮助我们做数学，让我们做以前没有时间做的事情。”

This is perhaps a paradigmatic example of how AI can be useful today.

这也许是人工智能在当今如何发挥作用的一个典型例子。

A group of expert mathematicians, with help from a big tech company, figures something out faster than they likely would have otherwise — and they are sure it is correct, because they can check it line by line.

一群专家数学家在一家大型科技公司的帮助下，以比其他方式更快的速度计算出一些结果，并且他们确信它是正确的，因为他们可以逐行检查它。

In asking what AI is doing to mathematical research, we shouldn’t only look at the successes.

在询问人工智能对数学研究有何影响时，我们不应只关注其成功。

Litt cautioned that “there is a lot of pollution of the commons by AI-generated nonsense.” Joel David Hamkins of the University of Notre Dame said he is “despairing of this ocean of slop that is overwhelming our journal systems.”

利特警告说，“人工智能产生的废话对公共资源造成了很大的污染。”圣母大学的乔尔·戴维·哈姆金斯 (Joel David Hamkins) 表示，他“对淹没我们期刊系统的污水海洋感到绝望”。

Mathematicians are pinning their hopes on formal proof as the way to navigate this ocean of slop.

数学家们把希望寄托在形式证明上，以此作为驾驭这片泥水海洋的方法。

They’re converting proofs into a language that computers can understand, and then using computer programs to verify that all the logic in the proof pans out.

他们将证明转换为计算机可以理解的语言，然后使用计算机程序来验证证明中的所有逻辑是否正确。

“AI without validation is too unreliable to be of use in any serious application,” Tao said.

“未经验证的人工智能太不可靠，无法用于任何严肃的应用程序，”陶说。

“It’s very likely that this technology is bigger than the computer ,” Daniel Litt wrote in a recent analysis of AI’s potential impact on mathematics.

“这项技术很可能比计算机更强大，”丹尼尔·利特在最近分析人工智能对数学的潜在影响时写道。

Currently, formalizing mathematical proofs in this way is a time-consuming, intricate process that itself takes substantial mathematical knowledge and is a bit of a craft.

目前，以这种方式形式化数学证明是一个耗时且复杂的过程，本身需要大量的数学知识，而且有点技巧。

And so mathematicians are increasingly turning to “autoformalization,” in which AI models translate mathematical statements into formal, logical ones and then prove them.

因此，数学家越来越多地转向“自动形式化”，即人工智能模型将数学陈述转化为形式化的逻辑陈述，然后证明它们。

“For the first time,” Tao said, “it does feel like we could formalize a significant fraction of mathematics through AI.”

“第一次，”陶说，“感觉我们确实可以通过人工智能形式化数学的很大一部分。”

The other major challenge that many mathematicians see as a consequence of AI’s increasing ability to do math is how it will affect the way students learn.

许多数学家认为，人工智能数学能力不断增强带来的另一个主要挑战是，它将如何影响学生的学习方式。

Even the most ardent proponents of AI are concerned.

即使是人工智能最热心的支持者也感到担忧。

Ken Ono, a professor at the University of Virginia who recently took a leave of absence to become the “founding mathematician” at Axiom, told me he sees “a rosy picture about how AI can help mathematics research, but I am deeply concerned about the role of AI in the future of work and training at all levels.”

弗吉尼亚大学教授小野肯 (Ken Ono) 最近请假成为了 Axiom 的“创始数学家”，他告诉我，他看到了“人工智能如何帮助数学研究的美好前景，但我对人工智能在未来工作和各级培训中的作用深感担忧。”

Tao said, “Many of the problems we assign, AI can solve instantly.

陶说：“我们布置的许多问题，人工智能都可以立即解决。

This can discourage a lot of the students from building up their mental muscles.”

这可能会阻碍很多学生增强心理素质。”

Hamkins agreed.

哈姆金斯同意了。

“I used to assign quite a bit of homework.

“我曾经布置过很多作业。

I just can’t do it anymore,” he said; a substantial fraction of the assignments students turn in are written by AI.

我实在做不到了，”他说。学生提交的作业中有很大一部分是由人工智能撰写的。

“I don’t want to read it.

“我不想读它。

I don’t want to be the AI cop.” Though homework was highly pedagogically valuable, now “everything has to be in-class quizzes and work.

我不想成为人工智能警察。”尽管家庭作业在教学上非常有价值，但现在“一切都必须是课堂测验和作业。

It’s a problem for the entire academic profession.”

这是整个学术界面临的问题。”

As another mathematician at a leading research university told me, “There is a serious risk that, in parallel with accelerating the progress of serious mathematical researchers, AI prevents us from making more mathematical researchers.”

正如一所领先研究型大学的另一位数学家告诉我的那样，“人工智能在加速严肃数学研究人员进步的同时，也阻碍了我们培养更多数学研究人员，这是一个严重的风险。”

T he use of AI to conduct math research is quickly becoming routine, and will, if current trends continue, soon be no more noteworthy than the use of the LaTeX typesetting language, which mathematicians use to format technical expressions.

使用人工智能进行数学研究正在迅速成为常规，并且如果当前趋势继续下去，很快将不会比使用 LaTeX 排版语言更值得注意，数学家用 LaTeX 排版语言来格式化技术表达式。

Aside from the results discussed in this article, there have been dozens of others in recent months.

除了本文讨论的结果之外，近几个月还出现了数十个其他结果。

Norms are still emerging about how to account for AI contributions.

关于如何解释人工智能贡献的规范仍在不断涌现。

Some papers include detailed supplemental information about how human mathematicians interacted with LLMs, including transcripts of conversations.

一些论文包含有关人类数学家如何与法学硕士互动的详细补充信息，包括对话记录。

Some mention an AI’s contribution prominently in the abstract; others make only brief mention of an AI’s assistance in the acknowledgments.

有些人在摘要中突出地提到了人工智能的贡献；其他人仅在致谢中简单提及人工智能的帮助。

Some mathematicians take pains to point out that though AI helped with the research, they wrote the paper themselves; others credit AI with the writing as well.

一些数学家煞费苦心地指出，虽然人工智能帮助了这项研究，但论文是他们自己写的；其他人也将写作归功于人工智能。

Even with the rapid changes of the past year, none of the mathematicians I spoke to in reporting this piece fear that the subject will become obsolete.

尽管过去一年发生了迅速的变化，但在报道这篇文章时与我交谈过的数学家中没有一个担心这个主题会过时。

Tao gave the analogy of mathematicians trying to climb “a big mountain range with lots of tall mountains and lots of foothills.” Humans can only climb one step at a time, but they can plan a route to the top of a mountain like Everest.

陶给出了数学家试图攀登“一座有许多高山和山麓的大山脉”的比喻。人类一次只能攀登一个台阶，但他们可以规划一条通往珠穆朗玛峰等山顶的路线。

Meanwhile, Tao said, current AIs are like jumping robots.

与此同时，陶说，当前的人工智能就像跳跃机器人。

They can sometimes “parkour their way to the top of a 6-foot wall” that a human couldn’t climb.

它们有时可以“跑到人类无法攀爬的 6 英尺高的墙顶”。

But they can’t do long-term strategic planning.

但他们无法做长期战略规划。

Those 6 feet might become 10 feet, or 100, Tao imagines, but “the little jumping robots are nowhere near the Mount Everests of math.”

陶想象，这 6 英尺可能会变成 10 英尺，或者 100 英尺，但“这些小型跳跃机器人距离数学上的珠穆朗玛峰还很远。”

Pak thinks that certain Everests — such as a major problem in number theory about whether sums like π + e can be written as fractions — will remain unresolved for centuries.

Pak 认为，某些珠穆朗玛峰——例如数论中关于 π + e 这样的和是否可以写成分数的一个主要问题——将在几个世纪内仍未得到解决。

“I’m really doubtful AI can make any dents there at all,” he said.

“我真的很怀疑人工智能能否在这方面产生任何影响，”他说。

“This is not something that AI would be able to do.

“这不是人工智能能够做到的事情。

But I’m quite positive that if humanity survives, eventually we will figure it out.”

但我非常肯定，如果人类能够生存下去，我们最终会找到答案。”

Of course, a lot depends on how the capabilities of AI algorithms change and improve in coming years.

当然，这在很大程度上取决于人工智能算法的能力在未来几年如何变化和改进。

Even the most astute and careful observers can’t say for sure how the models will develop.

即使是最精明、最细心的观察者也无法确定模型将如何发展。

Few see signs of stagnation.

很少有人看到停滞的迹象。

“Things are moving very fast.

“事情进展得非常快。

I don’t see any sign they are slowing down,” Litt said.

我没有看到任何迹象表明他们正在放缓，”利特说。

The first few months of 2026 have already seen a steady stream of new results from big companies like Google and OpenAI and small ones like Axiom , as well as from academics and even hobbyists .

2026 年前几个月，谷歌和 OpenAI 等大公司、Axiom 等小公司以及学者甚至业余爱好者已经源源不断地获得了新成果。

“My expectation is surely in 20 years we are going to see AI tools generating mathematics that in many measurable ways are better than every human mathematician,” Litt said.

“我的期望肯定是，20 年后，我们将看到人工智能工具生成的数学在许多可衡量的方面都比每个人类数学家都要好，”利特说。

“I would be shocked if that doesn’t happen.”

“如果这没有发生，我会感到震惊。”

But as Venkatesh told me, “In the end, there are infinitely many ways to formulate any piece of math.” The choices we make, he said, are governed by human values and shaped by the fact that mathematics is not only a science but also an art.

但正如 Venkatesh 告诉我的那样，“最终，表达任何数学的方法都有无限多种。”他说，我们做出的选择受人类价值观的支配，并受到数学不仅是一门科学而且是一门艺术这一事实的影响。

That balance between science and art is in large measure what gives math its beauty — one of the “valuable things in our culture” that Venkatesh wants to retain.

科学与艺术之间的平衡在很大程度上赋予了数学之美——这是文卡特什希望保留的“我们文化中有价值的东西”之一。

If AI pushes mathematics away from its artistic heritage, the discipline will be diminished, even if more theorems are proved each month.

如果人工智能将数学推离其艺术遗产，即使每个月都会证明更多的定理，这门学科也会被削弱。

After all, no poet talks seriously about doing statistical regression on sonnets to find the optimal ones.

毕竟，没有诗人认真谈论对十四行诗进行统计回归以找到最佳十四行诗。

The best hope for AI is that it will help mathematicians find and prove things that would otherwise have remained mysteries.

人工智能最大的希望是，它将帮助数学家发现并证明一些原本仍是谜团的事物。

Most mathematicians agree that that’s what computers have done for the past 80 years.

大多数数学家都认为这就是计算机在过去 80 年里所做的事情。

But the scale of the change now underway has left many feeling unsettled.

但目前正在发生的变革规模让许多人感到不安。

The biggest annual mathematics conference in the world is held every year in early January.

世界上最大的年度数学会议于每年一月初举行。

In 2026, in Washington, D.C., nervous jokes about being made obsolete by AI were plentiful, even if, on the record, everyone insisted that AI will be a helpmate to human mathematicians.

2026 年，在华盛顿特区，关于人工智能将被淘汰的紧张笑话比比皆是，尽管在记录中，每个人都坚持认为人工智能将成为人类数学家的帮手。

Williamson — who has been working with AI for years and is very excited by it — was chosen to deliver a series of prestigious lectures about AI and math to the entire conference.

威廉姆森多年来一直致力于人工智能领域，对此感到非常兴奋，他被选为整个会议发表一系列有关人工智能和数学的著名讲座。

He told the audience that it’s a mistake to react to AI developments with ignorance and fear.

他告诉观众，以无知和恐惧来应对人工智能的发展是错误的。

But he said he understands where the fear comes from.

但他说他理解恐惧从何而来。

He sees mathematics as a “craft that people have spent their lives — dedicated their lives — towards.

他将数学视为“人们花费一生——奉献一生——的一门手艺。

There is some possibility that its value may be greatly diminished in the future.”

未来它的价值可能会大幅缩水。”

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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。