How Terry Tao Became an Evangelist for AI in Math
Automated proof-checkers such as Lean can provide ironclad assurances that mathematical proofs are valid.
像 Lean 这样的自动校对器可以提供数学证明有效的铁定保证。
Samuel Velasco/ Quanta Magazine.
塞缪尔·贝拉斯科/广达杂志。
Code courtesy of Alex Kontorovich
代码由 Alex Kontorovich 提供
The following has been adapted from The Proof in the Code: How a Truth Machine Is Transforming Math and AI by Kevin Hartnett.
以下内容改编自 Kevin Hartnett 的《代码中的证明：真理机器如何改变数学和人工智能》。
T erry Tao has never been afraid of unconventional ideas.
陶哲轩从不害怕非传统的想法。
In November 2014, he was on a panel of five distinguished mathematicians, all inaugural recipients of the Breakthrough Prize in Mathematics, which came with a $3 million award.
2014 年 11 月，他是由五位杰出数学家组成的小组成员之一，他们都是数学突破奖的首届获得者，该奖项的奖金为 300 万美元。
The laureates’ conversation ranged from whether mathematics is invented or discovered — most of the mathematicians agreed that, at the very least, it feels like an act of discovery — to an assessment of the odds that we’re living in a digital simulation.
获奖者的讨论范围从数学是发明还是发现——大多数数学家都认为，至少感觉像是一种发现的行为——到对我们生活在数字模拟中的可能性的评估。
“Yeah, I think we’re actually not real,” said Maxim Kontsevich, who did his most important work in the 1990s at the intersection of math and physics.
“是的，我认为我们实际上并不真实，”马克西姆·康采维奇 (Maxim Kontsevich) 说道，他在 20 世纪 90 年代在数学和物理的交叉领域做了最重要的工作。
The computer program Lean might revolutionize mathematics.
计算机程序精益可能会彻底改变数学。
It has verified more than a quarter of a million theorems and led to new insights.
它已经验证了超过一百万个定理中的四分之一，并带来了新的见解。
But does it change what math is, or can be?
但它是否改变了数学是什么，或者可以改变什么？
Yet over the course of the 40-minute discussion, the statements that drew the most incredulity were Tao’s.
然而，在40分钟的讨论过程中，最令人怀疑的是陶的言论。
He predicted that in the future, instead of working alone or in small teams of two or three, mathematicians might work on projects with hundreds of other people at a time.
他预测，在未来，数学家可能会与数百人同时合作开展项目，而不是单独工作或两三人组成的小团队。
And when these collaborations were over, he said — in his modest, understated way — the results might be checked not by human referees but by computers.
当这些合作结束时，他以谦虚、低调的方式表示，结果可能不是由人类裁判而是由计算机来检查。
“One day we may actually write our papers not in LaTeX, but in some language which some smart software will convert to a formal language, and every so often you’ll get a compilation error — the computer does not understand how you derived this step,” he said.
“有一天，我们实际上可能不是用 LaTeX 写论文，而是用某种语言写论文，某些智能软件会将其转换为正式语言，时不时你会遇到编译错误 - 计算机不理解你是如何导出这一步的，”他说。
The statement was greeted by the event moderator and the other laureates as preposterous enough to make the simulation hypothesis seem reasonable by comparison.
活动主持人和其他获奖者对这一声明表示欢迎，认为其荒谬程度足以使模拟假设相比之下显得合理。
Even more surprising than the idea of hundreds of mathematicians working together was the fact that such a collaboration would appeal to Tao — because if anyone in the world seemed well suited to going it alone, it was him.
比数百名数学家一起工作的想法更令人惊讶的是，这样的合作对陶来说很有吸引力——因为如果世界上有人看起来很适合单独行动，那就是他。
The Proof in the Code explores the age of human-computer mathematical collaboration.
代码中的证明探索了人机数学协作的时代。
Samuel Velasco/ Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
Tao was born in 1975 in Adelaide, Australia, three years after his parents immigrated to the country from Hong Kong.
陶1975年出生于澳大利亚阿德莱德，三年前他的父母从香港移民到澳大利亚。
The first signs that their firstborn son was different came early.
他们的长子与众不同的第一个迹象很早就出现了。
When Tao was 2 and his family was visiting friends, his parents found him gathered with several 6-year-olds, demonstrating how to count using wooden blocks.
当陶两岁的时候，他的家人去拜访朋友，他的父母发现他和几个六岁的孩子聚集在一起，演示如何使用木块数数。
Asked how he’d learned to count things, he responded that he had seen it on Sesame Street .
当被问及他是如何学会数数时，他回答说他在《芝麻街》上看到过。
Five years later, when Tao was 7, he began learning calculus.
五年后，当陶七岁的时候，他开始学习微积分。
For three weeks in the spring of 1985, Tao’s parents brought him to the United States, where he met with Julian Stanley, director of the Study of Mathematically Precocious Youth, then at Johns Hopkins University.
1985年春天，陶的父母带他去了美国三个星期，在那里他遇到了当时在约翰·霍普金斯大学数学早熟青少年研究中心主任朱利安·斯坦利。
Stanley described Tao as having the greatest mathematical ability he had ever seen.
斯坦利形容陶拥有他所见过的最伟大的数学能力。
That same year Tao met the acclaimed mathematician Paul Erdős during the latter’s visit to Adelaide.
同年，陶渊明在著名数学家保罗·埃尔多斯 (Paul Erdős) 访问阿德莱德期间结识了后者。
A famous picture shows the grandfatherly Erdős, 72 at the time, reading a document in his lap while Tao, 10 years old with thick black hair, looks on intently, fingers raised thoughtfully to his chin.
一张著名的照片显示，当时 72 岁的祖父埃尔多斯 (Erdős) 坐在腿上阅读一份文件，而 10 岁的陶，一头浓密的黑发，聚精会神地看着，手指若有所思地举到下巴上。
Tao’s young legend grew when he entered the International Math Olympiad in 1986.
1986年，陶涛参加国际数学奥林匹克竞赛，他的年轻传奇就此开始。
He won a bronze medal that first year, becoming, at the age of 10, the youngest competitor ever to achieve that result.
第一年，他就获得了铜牌，年仅 10 岁，成为有史以来获得这一成绩的最年轻的参赛者。
In the two succeeding years he became the youngest-ever silver medalist and finally the youngest person ever to win a gold medal.
在接下来的两年里，他成为有史以来最年轻的银牌获得者，并最终成为有史以来最年轻的金牌获得者。
His formal education proceeded at a similarly accelerated pace.
他的正规教育也以同样的加快速度进行。
He graduated from the local Flinders University in Adelaide when he was 15 and, in the fall of 1992, boarded a plane with his father for New Jersey, where he started a Ph.D. in math at Princeton University.
15 岁时，他从阿德莱德当地的弗林德斯大学毕业，1992 年秋天，与父亲一起登上飞往新泽西州的飞机，在那里他开始攻读博士学位。普林斯顿大学数学博士。
Erdős had endorsed Tao’s early admission to the program, writing in a letter of recommendation, “I am sure he will develop into a first-rate mathematician and perhaps into a really great one.”
埃尔多斯对陶涛的早期录取表示认可，并在一封推荐信中写道：“我相信他会发展成为一名一流的数学家，甚至可能成为一名真正伟大的数学家。”
Paul Erdős, 72 at the time, with a 10-year-old Terrence Tao.
Paul Erdős，当时 72 岁，和 10 岁的 Terrence Tao。
Erdős was right.
埃尔多斯是对的。
By the time Tao was 24, he had made enough new discoveries to have his choice of permanent faculty positions; he ultimately decided to settle at the University of California, Los Angeles.
到 24 岁的时候，陶已经有了足够多的新发现，可以选择担任永久教职。他最终决定在加州大学洛杉矶分校定居。
Around that time, he met a young English number theorist named Ben Green.
大约在那个时候，他遇到了一位年轻的英国数论学家本·格林。
The two began collaborating on a proof that certain kinds of patterns called arithmetic progressions — in which the numbers in a set increase by a fixed interval, like 7, 10, 13, 16 — inevitably appear in large collections of prime numbers, despite the fact that primes appear to be scattered randomly along the number line.
两人开始合作证明某些称为算术级数的模式（其中一组数字按固定间隔增加，如 7、10、13、16）不可避免地出现在大量素数集合中，尽管素数似乎沿着数轴随机分散。
Their proof would become the signature result of Tao’s early career, contributing to his winning the Fields Medal in 2006, and propelling him to the upper echelons of mathematics.
他们的证明将成为陶渊明早期职业生涯的标志性成果，为他赢得 2006 年菲尔兹奖做出了贡献，并将他推向了数学的高层。
T ao could have built a successful career without collaborating with anyone, but that’s not the way he liked to work.
陶本可以在不与任何人合作的情况下建立成功的职业生涯，但这不是他喜欢的工作方式。
He viewed working with other researchers as a primary way to discover new ideas — take what you know, pair it with what I know, and see what happens.
他认为与其他研究人员合作是发现新想法的主要方式——将你所知道的与我所知道的结合起来，看看会发生什么。
This approach led Tao’s mathematical research to range over an unusually broad set of topics, from analytic number theory, including the Green-Tao theorem about prime numbers, to analysis, where he studied properties of the Navier-Stokes equations that describe the behavior of fluids, to algorithms for constructing MRI images from digital data.
这种方法使陶的数学研究涵盖了异常广泛的主题，从解析数论（包括关于素数的格林-陶定理）到分析（他研究了描述流体行为的纳维-斯托克斯方程的性质），再到从数字数据构建 MRI 图像的算法。
(The MRI collaboration developed during conversations Tao had with Emmanuel Candès, a statistician then at the California Institute of Technology, while they were both dropping off their kids at preschool.) This thirst for collaborative discovery also led Tao to do a lot of his work in public.
（MRI 合作是在陶和当时在加州理工学院担任统计学家的 Emmanuel Candès 的谈话中发展起来的，当时他们都在送孩子上学前班。）这种对合作发现的渴望也促使陶在公共场合做了很多工作。
In 2007, he started a blog , where he began publishing regular updates about his research.
2007 年，他开设了一个博客，开始定期发布有关他的研究的最新动态。
By that point, Tao was one of the most famous mathematicians not only in his field but in the world.
那时，陶不仅是他的领域，而且是世界上最著名的数学家之一。
His posts received a lot of attention and sometimes led to long exchanges in the comments section, where Tao enthusiastically participated.
他的帖子受到了很多关注，有时会在评论区引发长时间的交流，陶先生也热情参与其中。
He did it because he found it fun, and because he hoped the conversation might generate new ideas.
他这样做是因为他觉得这很有趣，并且因为他希望谈话能够产生新的想法。
Around that time, another early math blogger had a similar thought.
大约在那个时候，另一位早期的数学博主也有类似的想法。
Like Tao, Timothy Gowers was a prominent research mathematician with a taste for public exchange.
和陶一样，蒂莫西·高尔斯也是一位杰出的研究数学家，热衷于公共交流。
But rather than trusting serendipity to strike in his blog’s comment section, Gowers wanted to channel public energy in a focused way.
但高尔斯不想相信博客评论部分会出现偶然事件，而是希望以集中的方式引导公众能量。
In January 2009, he published a blog post announcing his desire to facilitate a new kind of “massively collaborative mathematics.” He would propose a problem in an open online forum, and “anybody who had anything whatsoever to say about the problem could chip in.” He named it the Polymath Project.
2009 年 1 月，他发表了一篇博客文章，宣布他希望促进一种新型的“大规模协作数学”。他会在一个开放的在线论坛上提出一个问题，“任何对这个问题有任何意见的人都可以参与进来”。他将其命名为博学者项目。
Tao viewed working with other researchers as a primary way to discover new ideas — take what you know, pair it with what I know, and see what happens.
陶认为与其他研究人员合作是发现新想法的主要方式——将你所知道的与我所知道的结合起来，看看会发生什么。
Tao jumped in.
涛跳了进去。
Like Gowers, he understood that some math problems were more amenable than others to being solved through large-scale collaboration.
和高尔斯一样，他明白有些数学问题比其他问题更适合通过大规模合作来解决。
The key, as Tao wrote in a comment on Gowers’ initial post, was to find problems that could “generate a number of simpler sub-problems …
正如陶在高尔斯最初的帖子的评论中所写，关键是找到可以“产生许多更简单的子问题......
which can largely be worked on in parallel.” By breaking big problems into individual cases, different teams or individuals could work on their own and then assemble their results as pieces of a bigger whole.
这在很大程度上可以并行进行。”通过将大问题分解为单独的案例，不同的团队或个人可以独立工作，然后将他们的结果组装成一个更大的整体。
At the same time, Tao knew that perhaps the biggest challenge with the Polymath model would be organizing: moderating contributions and checking to make sure that all the contributions were correct.
与此同时，陶知道，Polymath 模型的最大挑战可能是组织：调整贡献并检查以确保所有贡献都是正确的。
For the first Polymath project, Gowers proposed improving a result called the Hales-Jewett theorem, which was about patterns that appear when you shade cells in a grid with one of two different colors.
对于第一个 Polymath 项目，高尔斯提出改进称为 Hales-Jewett 定理的结果，该定理涉及当您用两种不同颜色之一对网格中的单元格进行着色时出现的模式。
After a few months of work, coordinated through thousands of comments by dozens of mathematicians, the group had proved a more exact statement about how those coloring patterns emerge.
经过几个月的工作，通过数十名数学家的数千条评论的协调，该小组已经证明了关于这些着色模式如何出现的更准确的陈述。
That fall, they released the work as a first-of-its-kind math paper with the pseudonymous byline “D.H.J.
那年秋天，他们发布了这篇论文，作为第一篇数学论文，其笔名是“D.H.J.”。
Polymath.” Gowers’ experiment had been a success.
博学者。”高尔斯的实验取得了成功。
It allowed many mathematicians — professional and amateur alike — to work together and yielded a proof in the end.
它让许多数学家——无论是专业的还是业余的——一起工作并最终得出了证明。
Over the next decade, there were 15 more Polymath projects, some of which Tao led, and the initiative attracted mainstream attention.
在接下来的十年里，又出现了 15 个 Polymath 项目，其中一些是由陶领导的，这一举措引起了主流关注。
On October 29, 2011, The Wall Street Journal ran an article called “The New Einsteins Will Be Scientists Who Share” and reported that the Polymath Project had “pioneered a new approach to problem-solving.”
2011 年 10 月 29 日，《华尔街日报》发表了一篇题为“新爱因斯坦将成为善于分享的科学家”的文章，报道博学者项目“开创了一种解决问题的新方法”。
Samuel Velasco/ Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
Yet in other ways, the Polymath Project was an idea before its time.
但从其他方面来说，博学者项目是一个超前的想法。
Tao found it exhilarating to be at the center of a frenzy of mathematical activity, but he recognized that the comments section of a blog was a limited platform for collaboration.
陶发现成为狂热数学活动的中心令人兴奋，但他认识到博客的评论部分是一个有限的协作平台。
Massive open collaboration increased the likelihood of a certain kind of serendipitous discovery, but at the same time it heightened the odds that any one of the many participants would contribute a mistake.
大规模的开放合作增加了某种偶然发现的可能性，但同时也增加了众多参与者中任何一个犯错的可能性。
The only way to guard against error was for a moderator to carefully check all the work.
防止错误的唯一方法是由主持人仔细检查所有工作。
But that kind of moderation bottleneck undermined the Polymath vision.
但这种节制瓶颈破坏了博学者的愿景。
What Tao was really after was an efficient new form of scientific discovery.
道真正追求的是一种有效的新形式的科学发现。
And after a while, he came to understand that the Polymath model was not it.
过了一段时间，他开始明白博学者模型并不是这样的。
To make it real, he thought, some kind of computer verification would be needed — a way to check contributions automatically, rather than by hand.
他认为，为了实现这一目标，需要某种计算机验证——一种自动检查贡献的方法，而不是手工检查。
But given the state of technology in the 2010s, he might as well have wished for passenger service to Mars.
但考虑到 2010 年代的技术水平，他可能还希望有前往火星的客运服务。
T ao had been aware of computer-verified mathematics for years.
陶多年来一直了解计算机验证的数学。
He knew about a few success stories, but he also knew that formal math was still impractical, requiring far more effort than it was worth in most cases.
他知道一些成功的故事，但他也知道正式的数学仍然不切实际，在大多数情况下需要付出比其价值更多的努力。
Nevertheless, Tao was intrigued by its potential.
尽管如此，陶对其潜力很感兴趣。
Almost uniquely among the world’s elite mathematicians, he saw the potential in new methods for doing mathematics.
在世界精英数学家中，他几乎是独一无二的，他看到了数学新方法的潜力。
In July 2022, in part to satisfy his curiosity, he began to organize a workshop on all the different ways computers were assisting mathematical research.
2022 年 7 月，部分为了满足他的好奇心，他开始组织一个研讨会，讨论计算机协助数学研究的所有不同方式。
He brought on a team of co-organizers, including Kevin Buzzard, who was at the time the world’s most visible evangelist for formal mathematics.
他召集了一个联合组织者团队，其中包括当时世界上最著名的形式数学传播者凯文·巴扎德 (Kevin Buzzard)。
Going into the conference, Tao regarded Lean, software that allows mathematical proofs to be written and checked as computer code, as a complicated program that would take months to learn.
在参加会议时，陶认为精益是一种允许以计算机代码形式编写和检查数学证明的软件，是一个需要数月才能学习的复杂程序。
Buzzard convinced him to give it a try.
巴扎德说服他尝试一下。
Along with that encouragement, Tao felt a strong responsibility to lead by example — if he was going to continue promoting machine-assisted proof, he needed to start trying it himself.
伴随着这种鼓励，陶感到自己有强烈的责任以身作则——如果他要继续推广机器辅助证明，他需要开始自己尝试。
On October 9, 2023, Tao posted on social media, “I have decided to finally get acquainted with the #Lean4 interactive proof system (using AI assistance as necessary to help me use it).”
2023年10月9日，陶在社交媒体上发帖称，“我决定终于熟悉#Lean4交互式证明系统（必要时使用人工智能辅助来帮助我使用它）。”
Tao felt a strong responsibility to lead by example — if he was going to continue promoting machine-assisted proof, he needed to start trying it himself.
陶感到以身作则的强烈责任——如果他要继续推广机器辅助证明，他需要自己开始尝试。
On MathOverflow, a popular online discussion forum for mathematicians, Tao found a question about something called Maclaurin’s inequality.
在MathOverflow（一个流行的数学家在线讨论论坛）上，陶发现了一个关于麦克劳林不等式的问题。
He decided to answer it as an experiment in formalization.
他决定将其作为形式化实验来回答。
First, he wrote up the proof as a typical math paper.
首先，他将证明写成一篇典型的数学论文。
It was short, only 10 pages long.
它很短，只有10页长。
Then he turned his attention to his real goal: seeing if he could formalize the simple proof in Lean.
然后他将注意力转向他的真正目标：看看他是否可以在精益中形式化这个简单的证明。
Initially, Tao thought he might be able to do it in a week, but he was quickly confronted by the differences between writing math by hand and typing it in Lean.
最初，Tao 认为他也许可以在一周内完成，但他很快就遇到了手写数学和用 Lean 打字之间的差异。
Tao observed that the hard parts of the proof were easy to formalize in Lean, while the simple parts took a surprising amount of work.
陶观察到，证明的困难部分很容易在精益中形式化，而简单的部分则需要大量的工作。
In the regular paper, Tao spent no time at all asserting that if you have three numbers, all of which are greater than 1, their sum is necessarily at least 3.
在普通论文中，陶根本没有花时间断言，如果你有三个数字，并且都大于 1，那么它们的总和必然至少为 3。
But Lean doesn’t abide assertions, and Tao had to spend time digging up a lemma in Mathlib — a digital library of already-formalized mathematics that Lean users draw on when writing proofs — that proved the self-evident relationship.
但 Lean 不遵守断言，Tao 不得不花时间在 Mathlib 中挖掘引理——一个已经形式化的数学数字图书馆，Lean 用户在编写证明时会使用它——证明了不言而喻的关系。
Similarly, in informal math, it’s not necessary to always specify which number system you’re working in.
同样，在非正式数学中，没有必要总是指定您正在使用的数字系统。
The number 3, for example, is simultaneously an integer, a natural number, and a real number.
例如，数字3同时是整数、自然数和实数。
In his original paper, Tao could simply write “3” without specifying what kind of 3 he had in mind.
在他的原始论文中，Tao 可以简单地写“3”，而没有具体说明他想要的 3 是什么样的。
However, in Lean he had to spell it out.
然而，在《精益》中，他必须把它拼出来。
Tao found his proof kept failing to compile because he had neglected to specify the correct type at different points in the formalization.
陶发现他的证明一直无法编译，因为他忽略了在形式化的不同点指定正确的类型。
It wasn’t until nearly a month later, on November 6, that Tao posted in a comment on his blog, “Just a remark that I have managed to formalize the results of this paper in Lean4.” The result was minor, and the Lean code he had written to formalize it was terrible.
直到近一个月后的11月6日，陶才在他的博客上发表评论，“只是说我已经成功地将这篇论文的结果在Lean4中形式化了。”结果很小，而他为形式化而编写的精益代码则很糟糕。
Yet Tao was now officially a contributing member of the Lean community.
然而，陶现在正式成为精益社区的贡献成员。
A t the same time he was learning Lean, Tao continued work on a host of other research projects.
在学习精益的同时，陶继续从事许多其他研究项目。
These included one with his longtime collaborators Ben Green and Tim Gowers, as well as Freddie Manners, a former student of Green’s and now a professor at the University of California, San Diego.
其中包括他的长期合作者本·格林（Ben Green）和蒂姆·高尔斯（Tim Gowers），以及格林的前学生、现任加州大学圣地亚哥分校教授弗雷迪·曼纳斯（Freddie Manners）。
It was an elite set of collaborators — Gowers, like Tao, had won a Fields Medal, while Green was among the most decorated number theorists in the field.
这是一群精英合作者——高尔斯和陶一样，赢得了菲尔兹奖，而格林是该领域获奖最多的数论学家之一。
The group had a particular problem in their sights, one that revolved around a mathematical object called a sumset.
该小组有一个特殊的问题，这个问题围绕着一个称为求和集的数学对象。
If you have a collection of numbers, it can be used to form another, related collection: its sumset.
如果你有一个数字集合，它可以用来形成另一个相关的集合：它的总和。
The sumset is made by taking the sum of every unique pair of numbers in the first set.
总和是通过将第一组中每对唯一的数字相加得到的。
All those sums together form the sumset of the original set.
所有这些总和共同构成了原始集合的总和。
If the original set is full of random numbers, then its sumset will be comparatively large.
如果原始集合充满了随机数，那么它的和就会比较大。
A set of 10 random numbers has a sumset of about 50 numbers (and a set of 1,000 numbers has a sumset of about 500,000 numbers).
一组 10 个随机数的总和约为 50 个数字（一组 1,000 个数字的总和约为 500,000 个数字）。
But if, instead of containing random numbers, the original set follows some kind of pattern, its sumset will be much smaller because many sums will appear multiple times (and you only include each sum once in the sumset).
但是，如果原始集合不包含随机数，而是遵循某种模式，则其总和将会小得多，因为许多总和会出现多次（并且您只在总和中包含每个总和一次）。
The set of the numbers 1 to 10 is one example — its sumset only contains 17 numbers (not 50, as you’d expect if it were a random collection of 10 numbers), because many of the sums repeat (1 + 6, 2 + 5, and 3 + 4 all equal 7, and you only enter 7 once in the sumset).
数字 1 到 10 的集合就是一个例子 - 它的总和只包含 17 个数字（不是 50，正如您所期望的，如果它是 10 个数字的随机集合），因为许多总和重复（1 + 6、2 + 5 和 3 + 4 都等于 7，并且您只需在总和中输入 7 一次）。
In addition to having a small sumset, the numbers 1 through 10 are an example of an arithmetic progression because they increase by a constant interval.
除了总和较小之外，数字 1 到 10 也是算术级数的示例，因为它们按恒定间隔增加。
A conjecture with roots in the 1960s by the computer scientist Katalin Marton asserts that this isn’t a coincidence.
计算机科学家 Katalin Marton 于 20 世纪 60 年代提出的猜想断言这并非巧合。
She predicted that sets that produce small sumsets must also include long arithmetic progressions.
她预测产生小总和的集合也必须包含长算术级数。
Gowers, Green, and Tao had made headway on a refined version of this problem called the polynomial Freiman-Ruzsa conjecture in the early 2000s but eventually got stuck.
高尔斯、格林和陶在 2000 年代初期在这个问题的改进版本（称为多项式 Freiman-Ruzsa 猜想）上取得了进展，但最终陷入了困境。
Then, in 2023, Tao, Green, and Manners picked it up again with an eye toward introducing techniques from probability theory that Manners had developed.
然后，在 2023 年，Tao、Green 和 Manners 再次拾起它，着眼于引入 Manners 开发的概率论技术。
They realized that by combining those techniques with Gowers’ earlier ideas, they might be able to solve the whole thing .
他们意识到，通过将这些技术与高尔斯早期的想法相结合，他们也许能够解决整个问题。
They brought Gowers into the collaboration, and the quartet made steady progress through the summer of 2023.
他们邀请高尔斯加入合作，四人组在 2023 年夏天取得了稳步进展。
By late fall, they had it.
到了深秋，他们做到了。
On November 9 — just three days after Tao uploaded his first formal Lean proof to GitHub — they uploaded their proof to arxiv.org .
11 月 9 日——Tao 将他的第一个正式精益证明上传到 GitHub 三天后——他们将证明上传到了 arxiv.org。
With Lean on his mind, Tao suggested to his three co-authors that they could try formalizing their paper.
考虑到精益，陶向他的三位合著者建议他们可以尝试将他们的论文形式化。
The work seemed like a good candidate for formalization both because it was an important result and because it relied on relatively simple techniques.
这项工作似乎是形式化的良好候选者，因为它是一个重要的结果，而且因为它依赖于相对简单的技术。
They wouldn’t have to spend months adding prerequisite material to Mathlib — most of the necessary definitions were already there.
他们不必花费数月时间向 Mathlib 添加必备材料 - 大多数必要的定义已经存在。
However, Green, Gowers, and Manners weren’t especially interested in taking the time to learn Lean.
然而，格林、高尔斯和曼纳斯对花时间学习精益并不是特别感兴趣。
So Tao set off on his own — though he knew he likely wouldn’t be alone for long.
于是，陶独自出发了——尽管他知道自己可能不会孤独太久。
Any project he led was likely to draw attention.
他领导的任何项目都可能引起关注。
On November 13, Tao kicked off a new channel in a Lean-focused chat group.
11 月 13 日，Tao 在一个专注于精益的聊天组中开通了一个新频道。
“Hi everyone.
“大家好。
I am thinking of starting a project to formalize in Lean4 the recent proof of Timothy Gowers, Ben Green, Freddie Manners, and myself of the polynomial Freiman-Ruzsa (PFR) conjecture,” he wrote.
我正在考虑启动一个项目，在 Lean4 中形式化 Timothy Gowers、Ben Green、Freddie Manners 和我自己最近对多项式 Freiman-Ruzsa (PFR) 猜想的证明，”他写道。
He would use the channel to coordinate activity on the project and would be “happy to accept volunteers to contribute to this project in whatever capacity they feel able.” It was a reboot of the Polymath Project, only this time they were formalizing an existing result rather than trying to prove a new one — and all the work would be verified by Lean, meaning Tao wouldn’t have to check it himself.
他将利用该渠道协调该项目的活动，并“很乐意接受志愿者以他们认为有能力的任何身份为该项目做出贡献。”这是博学项目的重启，只不过这一次他们将现有的结果形式化，而不是试图证明一个新的结果——所有的工作都将由精益验证，这意味着陶不需要亲自检查。
Equational Theories, in Tao’s view, was the opening act of what he hoped would become a new era of “experimental” mathematics.
在陶看来，方程理论是他希望成为“实验”数学新时代的开端。
Yaël Dillies, a graduate student at Cambridge University, soon set up a technical infrastructure for the project that divided the proof into 13 sections.
剑桥大学的研究生 Yaël Dillies 很快就为该项目建立了一个技术基础设施，将证明分为 13 个部分。
Within each section, Tao identified the sequence of lemmas and definitions that needed to be formalized.
在每个部分中，Tao 确定了需要形式化的引理和定义的顺序。
In a typical math paper, lemmas — simpler results that help build toward the proof of a larger theorem — might be about 20 lines long, but for the PFR formalization, Tao broke the proof down into five-line lemmas.
在典型的数学论文中，引理（帮助构建更大定理证明的更简单的结果）可能约为 20 行长，但对于 PFR 形式化，Tao 将证明分解为五行引理。
His goal was to make the proof as modular as possible, allowing many people to make small contributions.
他的目标是使证明尽可能模块化，让许多人做出小小的贡献。
For the first week, most of the activity on the thread was about formalizing basic concepts from probability theory that the proof required but were not yet in Mathlib.
第一周，线程上的大部分活动都是关于将概率论中的基本概念形式化，证明需要这些概念，但 Mathlib 中尚未包含这些概念。
In particular, they had to formalize Shannon entropy — a measure of the uncertainty or disorder in a data source, like a set of numbers.
特别是，他们必须将香农熵形式化——一种数据源中不确定性或无序性的度量，比如一组数字。
But along with formalizing math, Tao and the others spent that first week figuring out how to work together.
但除了将数学形式化之外，陶和其他人在第一周的时间里都在思考如何一起工作。
Initially, the conversation was free-form, with Tao posting about what he thought needed to be done and others chiming in with ideas about how to do it, much as the Polymath projects had unfolded in blog comments.
最初，对话是自由形式的，陶发表了他认为需要做的事情，其他人也提出了如何做的想法，就像博客评论中展开的博学项目一样。
On November 22, Tao posted a list of 22 outstanding lemmas and wrote, “If you want to claim one or more of these lemmas, please do so by replying in this thread.” The replies flooded in: “I’d like to claim the entropy of a uniform random variable :),” wrote Paul Lezeau, a Ph.D. student at the London School of Geometry and Number Theory.
11 月 22 日，Tao 发布了 22 个突出引理的列表，并写道：“如果您想声明其中一个或多个引理，请在此帖子中回复。”回复如潮水般涌来：“我想求一个均匀随机变量的熵:)，”博士保罗·勒索(Paul Lezeau)写道。伦敦几何与数论学院的学生。
“I’m gonna take a stab at the general fibring identity,” replied Aaron Anderson, a Ph.D. student at UCLA.
“我要尝试一下一般的纤维特性，”博士亚伦·安德森回答道。加州大学洛杉矶分校的学生。
Drawn by word of mouth, more and more mathematicians joined the effort.
经过口耳相传，越来越多的数学家加入了这一努力。
By the end of November, Tao, like a harried volunteer coordinator, was writing little Lean code himself, instead focusing on finding tasks for others to do.
到 11 月底，Tao 就像一位忙碌的志愿者协调员一样，自己编写了一些精益代码，而不是专注于为其他人寻找任务。
On November 28, he wrote, “Given that we may temporarily have a surplus of volunteers for the PFR project as it nears completion, I thought of one additional small task that someone might be willing to work on.” Forty-six minutes later, Kim Morrison replied that they had completed the task.
11 月 28 日，他写道：“鉴于 PFR 项目接近完成时我们可能暂时有多余的志愿者，我想到了一项额外的小任务，有人可能愿意去做。”四十六分钟后，金·莫里森回复说他们已经完成了任务。
“Wow, that was quick!
“哇哦，速度真快啊！
Thanks!” Tao answered.
谢谢！”陶回答道。
Even before the formalization was complete, the Lean community began discussing what it meant.
甚至在正式化完成之前，精益社区就开始讨论它的含义。
In particular, they debated whether the efficiency of the project signaled a new era of fast formalization, or whether it reflected the singular influence of Terry Tao.
他们特别争论了该项目的效率是否标志着快速正规化的新时代，或者是否反映了陶哲轩的独特影响力。
In a wrap-up post in the group thread, Tao reflected that he had not written much of the code himself.
在小组帖子的总结帖子中，Tao 表示，他自己编写的代码并不多。
“This is actually quite encouraging to me, as it suggests to me that it will be possible for mathematicians to lead Lean formalization projects without requiring extensive Lean programming skills (though one may need at least enough expertise to be able to state lemmas, if not prove them).” Eight minutes later, Johan Commelin, a mathematician at Utrecht University and the director of the Mathlib initiative, replied, “I don’t want to immediately hijack this thread,” before going on to question whether the lessons Tao had learned during the project were broadly applicable.
“这实际上对我来说非常令人鼓舞，因为它向我表明，数学家有可能在不需要广泛的精益编程技能的情况下领导精益形式化项目（尽管人们可能至少需要足够的专业知识来陈述引理，如果不是证明它们的话）。”八分钟后，乌得勒支大学数学家、Mathlib 计划的负责人 Johan Commelin 回答道：“我不想立即劫持这个线程”，然后继续质疑陶在该项目中学到的经验教训是否广泛适用。
“Of course you got a lot of help with this project because of its high-profile nature,” he wrote.
“当然，由于这个项目的高调性，你得到了很多帮助，”他写道。
Commelin also noted that while projects like PFR were fun and exciting to take part in, contributing to them was not the kind of thing young mathematicians were recognized for when they applied for academic jobs.
Commelin 还指出，虽然像 PFR 这样的项目参与起来既有趣又令人兴奋，但年轻数学家在申请学术工作时并不会因为为这些项目做出贡献而受到认可。
“At the moment, it is still not clear how formalizers (for lack of a better job description) will be credited by the mathematical community, and how these activities will be valued on the job market.” Tao replied, “For what it’s worth, I’m more than happy to mention contributions to this project in letters of reference as appropriate.”
“目前，尚不清楚数学界将如何评价形式化者（由于缺乏更好的职位描述），以及这些活动将如何在就业市场上受到重视。”陶回答说：“无论如何，我非常乐意在推荐信中酌情提及对这个项目的贡献。”
B y 2024, Tao had become the most prominent public voice touting the potential of machine-assisted mathematics.
到 2024 年，陶已经成为宣扬机器辅助数学潜力的最著名的公众声音。
He was three years into his tenure on President Joe Biden’s President’s Council of Advisors on Science and Technology and had become co-chair of a working group on generative AI.
他在乔·拜登总统的总统科学技术顾问委员会任职三年，并成为生成人工智能工作组的联合主席。
In a pair of high-profile speeches in 2024, he expressed his vision for a new kind of mathematical collaboration: one that combined human insight, the creativity of large language models, and the guarantees of correctness provided by formal verification systems.
在 2024 年的两次备受瞩目的演讲中，他表达了对一种新型数学协作的愿景：一种将人类洞察力、大型语言模型的创造力以及形式验证系统提供的正确性保证结合起来的协作方式。
He came to this view in part because he saw clear limits on what current AI tools could do.
他得出这一观点的部分原因是他看到当前人工智能工具的功能存在明显的局限性。
They excelled at solving straightforward problems or tasks with plenty of prior data, but at the frontier of mathematics — where there were few published results and little data to train on — AI faltered.
他们擅长利用大量先验数据解决简单的问题或任务，但在数学前沿——几乎没有发表的结果，也几乎没有可供训练的数据——人工智能却步履蹒跚。
In his early experiments with LLMs, he observed that they behaved like overconfident undergraduates, offering suggestions without the expertise to tell the difference between good and bad ideas.
在他对法学硕士的早期实验中，他观察到他们的行为就像过于自信的本科生，在没有专业知识来区分好想法和坏想法的情况下提供建议。
Yet Tao had a way forward in mind.
然而，陶心中有一条前进的道路。
He didn’t think AI would replace human mathematicians anytime soon, but he did consider it particularly well suited to helping solve certain types of complex mathematical problems: ones that could be broken into thousands of small, manageable subproblems — essentially the same class of problems that worked well for Polymath projects.
他并不认为人工智能会很快取代人类数学家，但他确实认为人工智能特别适合帮助解决某些类型的复杂数学问题：这些问题可以分解为数千个小的、可管理的子问题——本质上是适用于博学项目的同一类问题。
At that scale, mathematicians could employ AI to solve large swaths of the easiest subproblems, with its results outputted as formal proofs that Lean could check, and step in to handle the most difficult remaining questions themselves.
在这种规模上，数学家可以利用人工智能来解决大量最简单的子问题，其结果可以作为精益可以检查的正式证明输出，并自行处理最困难的剩余问题。
In 2024, Tao was promoting this vision to anyone who would listen, and following the PFR project, he had realized that if he really believed in the work, he needed to step up and lead it himself.
2024 年，Tao 向任何愿意倾听的人宣传这一愿景，在 PFR 项目之后，他意识到，如果他真的相信这项工作，他需要亲自站出来领导它。
He also knew right away which problem he would start with.
他也立即知道他将从哪个问题开始。
It was a question that he had stumbled upon a year earlier.
这是他一年前偶然发现的一个问题。
In July 2023, a user on MathOverflow posed a seemingly simple puzzle.
2023 年 7 月，MathOverflow 上的一位用户提出了一个看似简单的难题。
Consider an operation like addition, the user wrote.
用户写道，考虑一下像加法这样的操作。
It might follow certain fundamental algebraic laws like the commutative law, which says x + y = y + x , or the associative law, which states that ( x + y ) + z = x + ( y + z ).
它可能遵循某些基本代数定律，例如交换律，即 x + y = y + x ，或结合律，即 ( x + y ) + z = x + ( y + z )。
In many cases, there’s no relationship between one law and another — the commutative law doesn’t imply the associative law, for example.
在许多情况下，一个定律与另一个定律之间没有关系——例如，交换律并不意味着结合律。
The MathOverflow question concerned the relationship between two particular laws, and another user answered it quickly.
MathOverflow 问题涉及两个特定定律之间的关系，另一位用户很快就回答了。
But the question of how laws relate to each other in general caught Tao’s curiosity.
但法律之间一般如何相互关联的问题引起了陶的好奇。
Rather than solving puzzles one by one, Tao began sketching out a rough diagram that showed how different possible algebraic laws relate to one another.
陶没有一个一个地解决谜题，而是开始绘制一张粗略的图表，显示不同的可能代数定律如何相互关联。
It became clear the picture might be quite complicated.
很明显，情况可能相当复杂。
He saw that if he restricted his study to algebraic laws involving operations applied exactly four times, there were about 4,694 laws he had to account for.
他发现，如果他将研究限制在涉及运算恰好四次的代数定律上，那么他必须考虑大约 4,694 个定律。
Each law could potentially imply or fail to imply any other law, creating 22 million logical implications to check.
每条法律都可能暗示或无法暗示任何其他法律，从而产生 2200 万个需要检查的逻辑含义。
Once he had checked them all — either by proving they held or by finding a counterexample in which they failed — he would have a complete picture of how all 4,694 of those laws relate to each other.
一旦他检查了所有这些定律——无论是通过证明它们成立还是通过找到它们失败的反例——他就能全面了解所有 4,694 条定律是如何相互关联的。
It felt like exactly the right scale for the new style of mathematics he was proposing.
感觉这对于他所提出的新数学风格来说是完全正确的。
Tao called his new endeavor “ Equational Theories ” and announced its formation in a post on his personal blog on September 25, 2024.
陶将他的新尝试称为“方程理论”，并于 2024 年 9 月 25 日在其个人博客上发表文章宣布其成立。
He opened by ticking through the main reasons large-scale public math collaborations had been hard in the past and then wrote, “Proof assistant languages, such as Lean, provide a potential way to overcome these obstacles.”
他首先列举了过去大规模公共数学合作困难的主要原因，然后写道：“像 Lean 这样的证明辅助语言提供了克服这些障碍的潜在方法。”
To start, Tao and the growing number of volunteers who joined him tested the more than 4,000 laws against simple mathematical structures known as magmas.
首先，Tao 和越来越多的志愿者用称为岩浆的简单数学结构测试了 4,000 多个定律。
Magmas are stripped-down versions of arithmetic that made for a useful starting point because any law that failed to hold for magmas couldn’t possibly imply other, more complex laws.
岩浆是算术的简化版本，它是一个有用的起点，因为任何不适用于岩浆的定律不可能暗示其他更复杂的定律。
The participants quickly tested millions of these simplified systems using basic Python scripts and within days had resolved more than 99% of the 22 million potential implications.
参与者使用基本 Python 脚本快速测试了数百万个此类简化系统，并在几天内解决了 2200 万个潜在影响中的 99% 以上。
Tao posted on day 2 — September 27 — that he was astonished at how rapidly the project was advancing: “This project has moved far, far quicker, and scaled up much much faster, than I had expected — only 48 hours in and already a large fraction of the implications are likely to be resolved soon!
9 月 27 日，Tao 在第 2 天发帖称，他对该项目的进展速度感到惊讶：“这个项目进展得非常非常快，规模也比我预期的要快得多 — 仅用了 48 小时，而且很大一部分影响可能很快就会得到解决！
I thought the 3-week PFR project was fast, but this is an insane additional level of speed.”
我认为为期 3 周的 PFR 项目很快，但这是一个疯狂的额外速度水平。”
Once the simplest implications were resolved, the Equational Theories volunteers moved in a decentralized way to automated theorem provers that could search for solutions to problems all on their own, without interactive help.
一旦最简单的含义得到解决，方程理论志愿者就会以分散的方式转向自动化定理证明器，这些定理证明器可以自行寻找问题的解决方案，而无需交互式帮助。
These provers, along with old-fashioned human ingenuity, knocked down the open questions one at a time.
这些证明者，加上老式的人类聪明才智，一次一个地解决了悬而未决的问题。
Like a scientist watching his own creation come to life, Tao admired the work as it unfolded.
就像一位科学家看着自己的创造物变成现实一样，陶对这件作品的展开表示赞赏。
“The project seems to be successfully decentralizing; in particular, there is now a lot of activity going on now that I am not fully aware of,” he wrote.
他写道：“该项目似乎正在成功去中心化；特别是，现在正在进行很多我尚未完全意识到的活动。”
To many mathematicians, Tao’s project was intriguing but odd.
对于许多数学家来说，陶的项目很有趣，但也很奇怪。
Buzzard followed along, fascinated by the social experiment though bored by the mathematical content of Equational Theories.
巴扎德跟在后面，他对社会实验着迷，尽管对方程理论的数学内容感到厌倦。
He thought it was both elementary and weird, though he admired Tao’s inventiveness.
他认为这既简单又奇怪，尽管他很欣赏陶的创造力。
John Baez, another prominent mathematician, was more blunt.
另一位著名数学家约翰·贝兹 (John Baez) 则更为直言不讳。
He remarked, “This seems like a colossal waste of time to me,” before acknowledging that he felt the same way about college football, and plenty of people liked that, too.
他说，“这对我来说似乎是巨大的时间浪费”，然后他承认他对大学橄榄球也有同样的感觉，而且很多人也喜欢这一点。
Within a month, the Equational Theories group had narrowed 22 million questions down to 238.
一个月内，方程理论小组将 2200 万个问题缩减至 238 个。
By late November, they were down to 138.
到 11 月底，数量已降至 138 人。
As they chipped away at the remaining cases, progress slowed.
当他们逐步解决剩余的案件时，进展速度放慢了。
When the new year began, about 30 implications were left unresolved, and the rate of progress slowed even further.
新年伊始，大约有 30 个问题尚未解决，进展速度进一步放缓。
By the end of March, they had been stuck for several weeks on just four.
到三月底，他们已经被困在只有四颗星上好几个星期了。
Contributors tried their hand at the remainder, but with so few implications left, many people drifted away; Tao’s updates slackened from their near-daily cadence to once every few weeks.
贡献者尝试了剩下的部分，但由于留下的影响太少，许多人就离开了。陶的更新频率从几乎每天一次减少到每隔几周一次。
But settling every single one of the 22 million implications had never really been the goal of Equational Theories.
但解决 2200 万个影响中的每一个问题从来都不是方程理论的真正目标。
Out of sheer curiosity Tao had wanted a map of the complete landscape, and now he had that, minus a few details.
出于纯粹的好奇，陶想要一张完整的景观地图，现在他得到了，但减去了一些细节。
More importantly, he viewed Equational Theories as a pilot project for a fundamentally new way of doing mathematics — and in that regard it was an unqualified success.
更重要的是，他将方程理论视为一种全新的数学方法的试点项目——从这方面来说，它取得了绝对的成功。
Equational Theories, in Tao’s view, was the opening act of what he hoped would become a new era of “experimental” mathematics.
在陶看来，方程理论是他希望成为“实验”数学新时代的开端。
He had in mind the kind of transformation that had already come to fields like physics.
他想到的是物理学等领域已经发生的那种转变。
Physics had once been a largely theoretical discipline where solitary thinkers or small groups of collaborators tackled one or two problems at a time — in other words, it used to look a lot like math still did.
物理学曾经是一门主要是理论学科，孤独的思考者或一小群合作者一次解决一两个问题——换句话说，它过去看起来很像数学。
But with technological advancements came a new, experimental branch of the field — massive collaborations at laboratories like CERN’s Large Hadron Collider, where hundreds or even thousands of researchers with specialized skills worked together and generated huge volumes of data.
但随着技术的进步，该领域出现了一个新的实验性分支——在欧洲核子研究中心的大型强子对撞机等实验室进行大规模合作，数百甚至数千名具有专业技能的研究人员一起工作并生成大量数据。
These experiments didn’t replace theory but complemented it, with new results washing between the two modes of investigation.
这些实验并没有取代理论，而是对其进行了补充，新的结果在两种研究模式之间不断流动。
Tao imagined a similar evolution happening in mathematics.
陶想象数学中也会发生类似的演变。
He trusted that novel forms of inquiry would inevitably lead to novel insights, as they always had before.
他相信新颖的探究形式将不可避免地带来新颖的见解，就像以前一样。
As the Equational Theories team methodically crossed implications off their immense table, they stumbled onto genuinely new mathematical constructions — like “magma cohomology,” an alien extension of the concept of group cohomology, a deep and well-studied field describing when groups can or cannot be enlarged in certain ways.
当方程理论团队有条不紊地从他们巨大的表格中交叉含义时，他们偶然发现了真正新的数学结构——比如“岩浆上同调”，这是群上同调概念的外来扩展，这是一个深入且经过充分研究的领域，描述群何时可以或不可以以某种方式扩大。
Tao reached out to John Baez — the Equational Theories naysayer and an expert in cohomology — to ask if this construction had been seen before.
陶联系了约翰·贝兹（John Baez）——方程理论的反对者和上同调专家——询问这种构造以前是否见过。
Baez admitted he had never encountered it.
贝兹承认他从未遇到过这种情况。
To Tao, that was exactly the point.
对于陶来说，这正是重点。
The project had shown that mathematics could be conducted differently, experimentally — and in doing so, it had turned up something genuinely new.
该项目表明，数学可以通过不同的实验方式进行——并且在这样做的过程中，它发现了一些真正新的东西。
Tao had never expected Equational Theories to unearth a revelation; he wanted it to demonstrate the efficacy of a new kind of mathematical machine.
陶从来没有想到方程论会揭示出一个启示。他希望它能够证明一种新型数学机器的功效。
And in that sense it worked.
从这个意义上说，它起作用了。
Terry Tao had found a new way of doing mathematics, and he showed no signs of going back.
陶哲轩找到了一种新的数学方法，而且他没有任何回头的迹象。
Correction: September 21, 2026 An earlier version of this article misstated the contribution of Yaël Dillies and their affiliation at the time.
更正日期：2026 年 9 月 21 日 本文的早期版本错误地表述了 Yaël Dillies 的贡献及其当时的隶属关系。
Excerpted from The Proof in the Code: How a Truth Machine Is Transforming Math and AI by Kevin Hartnett.
摘自《代码中的证明：真理机器如何改变数学和人工智能》作者：Kevin Hartnett。
Copyright © 2026 by Kevin Hartnett.
版权所有 © 2026 凯文·哈内特。
To be published on June 9, 2026, by Quanta Books in partnership with Farrar, Straus and Giroux.
由 Quanta Books 与 Farrar、Straus 和 Giroux 合作于 2026 年 6 月 9 日出版。
All rights reserved.
版权所有。
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