In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far?
Kristina Armitage/ Quanta Magazine
克里斯蒂娜·阿米蒂奇/广达杂志
I n ancient Greece, Euclid showed that if you agree on a small list of preliminary principles, or axioms, you can use deductive reasoning to reveal all sorts of new mathematical truths.
在古希腊，欧几里得表明，如果你同意一小部分初步原理或公理，你就可以使用演绎推理来揭示各种新的数学真理。
But although these early proofs, as mathematicians call them, were derived using the laws of logic, they sometimes also contained hidden, unstated assumptions or relied on misleading intuitions.
但是，尽管这些早期的证明（正如数学家所说）是利用逻辑定律推导出来的，但它们有时也包含隐藏的、未陈述的假设或依赖于误导性的直觉。
In these cases, small gaps in a proof might go unnoticed; one couldn’t say with absolute confidence that it was correct.
在这些情况下，证明中的小缺陷可能会被忽视；人们不能绝对自信地说它是正确的。
Over the past few centuries, mathematicians have sought to close those gaps.
在过去的几个世纪里，数学家们一直在努力缩小这些差距。
By the early 1900s, they settled on the axioms they wanted to use.
到 1900 年代初，他们确定了想要使用的公理。
They also introduced different logical systems and standards in an effort to further “formalize” their arguments — to guarantee that, if you made every deduction in a proof explicit, its conclusion would have to be true no matter how long and convoluted it got.
他们还引入了不同的逻辑系统和标准，以进一步“形式化”他们的论点——以保证，如果你将证明中的每一个推论都明确化，那么无论它有多长、有多复杂，它的结论都必须是正确的。
The computer program Lean might revolutionize mathematics.
计算机程序精益可能会彻底改变数学。
But might it also erode conceptual and ideological diversity?
但这是否也会削弱概念和意识形态的多样性？
This formalization effort engendered trust.
这种正规化的努力产生了信任。
But it did far more than that.
但它所做的远不止于此。
The push to formalize has helped mathematicians uncover novel connections between different areas of math.
形式化的推动帮助数学家发现了不同数学领域之间的新联系。
It has taken the field in unexpected new directions.
它将该领域带向了意想不到的新方向。
It has taught us, said David Bressoud of Macalester College in Minnesota, that “you often don’t know what you don’t know.
明尼苏达州马卡莱斯特学院的戴维·布雷苏德 (David Bressoud) 表示，它告诉我们，“你常常不知道自己不知道什么。
And to be humble about that.”
并对此保持谦虚。”
But big advances in mathematics inevitably require bold ideas.
但数学的重大进步不可避免地需要大胆的想法。
These often come from experimentation and intuition — from exploring novel mathematical worlds and testing out novel theories, even if you can’t prove that every step in the journey logically follows from the last.
这些通常来自实验和直觉——来自探索新颖的数学世界和测试新颖的理论，即使你无法证明旅程中的每一步都逻辑上遵循最后一步。
It’s a less formal way of doing math that at first tends to contain mistakes.
这是一种不太正式的数学计算方式，一开始往往会出现错误。
It’s not easy to strike a fruitful balance between the creativity needed to paint new mathematical landscapes and the rigor necessary to ensure that the resulting pictures are true.
在描绘新的数学景观所需的创造力和确保所得到的图像真实性所需的严谨性之间取得富有成效的平衡并不容易。
Sometimes, formalization can upset that balance.
有时，形式化会破坏这种平衡。
What some mathematicians see as a move toward greater certainty, others might see as pedantry or a barrier to progress.
一些数学家认为这是迈向更大确定性的一步，而另一些数学家可能认为这是迂腐或进步的障碍。
Euclid’s Elements marked the first major effort to formalize mathematics in the West.
欧几里得的《几何原本》标志着西方数学形式化的首次重大努力。
It established a deductive method for proving statements that’s still used today.
它建立了一种至今仍在使用的证明陈述的演绎方法。
Above, pages from the first printed edition of the work, made in 1482.
上图是该书 1482 年出版的第一版印刷版的页面。
Folger Shakespeare Library/Creative Commons CC BY-SA 4.0
福尔格莎士比亚图书馆/知识共享 CC BY-SA 4.0
Today, mathematicians are attempting their most ambitious formalization project yet.
今天，数学家们正在尝试他们迄今为止最雄心勃勃的形式化项目。
They’re aiming to rewrite all of mathematics in a computer language known as Lean, which can then automatically verify their proofs.
他们的目标是用一种名为 Lean 的计算机语言重写所有数学，然后该语言可以自动验证他们的证明。
Writing a proof in Lean requires an enormous amount of time and effort, but so far, the program has been used to verify more than 260,000 theorems.
用 Lean 编写证明需要大量的时间和精力，但到目前为止，该程序已被用来验证超过 260,000 个定理。
It has the potential to put mathematics on the most solid foundation one can imagine.
它有潜力为数学奠定人们能想象到的最坚实的基础。
As with past attempts to formalize math, Lean has generated mixed feelings.
与过去将数学形式化的尝试一样，精益引发了复杂的感受。
Some mathematicians look forward to offloading tedious verification work onto computers and see Lean as a potentially transformative new way of doing mathematics.
一些数学家期待将繁琐的验证工作转移到计算机上，并将精益视为一种潜在的变革性数学新方法。
Others think that their time and resources are better spent elsewhere — or, worse, that a Lean-centric approach will distort the true value of mathematics.
其他人则认为他们的时间和资源最好花在其他地方，或者更糟糕的是，以精益为中心的方法会扭曲数学的真正价值。
It’s a discussion that’s starting to play out in the hallways of math departments around the world: How do we balance the creativity needed to discover new mathematical connections with the rigor needed to ensure that every logical step is undeniable?
这是一个开始在世界各地数学系走廊上展开的讨论：我们如何平衡发现新数学联系所需的创造力与确保每一个逻辑步骤都不可否认所需的严谨性？
A major milestone in the quest for rigor came about because of hidden problems with one of the most important mathematical achievements of all time.
追求严谨性的一个重要里程碑是由于有史以来最重要的数学成就之一的隐藏问题而出现的。
In the late 17th century, Isaac Newton and Gottfried Leibniz independently developed calculus, a technique for understanding how quickly a function changes at a given point.
17 世纪末，艾萨克·牛顿和戈特弗里德·莱布尼茨独立开发了微积分，这是一种了解函数在给定点变化速度的技术。
But in an informal sense, calculus had already been around for centuries.
但从非正式的意义上来说，微积分已经存在了几个世纪。
In fact, Archimedes was doing something like calculus in the third century BCE.
事实上，阿基米德在公元前三世纪就在做类似微积分的事情。
To calculate the area of a circle, he first studied the areas of shapes with straight sides called polygons.
为了计算圆的面积，他首先研究了称为多边形的直边形状的面积。
By using polygons with more and more sides, he was estimating a limit: the area of the circle.
通过使用边数越来越多的多边形，他估计了一个极限：圆的面积。
Newton and Leibniz took a similar approach, using limits to understand change.
牛顿和莱布尼茨采取了类似的方法，利用极限来理解变化。
In his treatise On Conoids and Spheroids , seen here in an 1897 translation, Archimedes introduced ideas that would play a key role in the development of calculus millennia later.
阿基米德在他的论文《论圆锥体和椭球体》（此处为 1897 年翻译）中引入了一些思想，这些思想将在数千年后微积分的发展中发挥关键作用。
From T.L.
来自 T.L.
Heath’s edition of The Woks of Archimedes.
希斯版的《阿基米德的锅》。
Cambridge University Press, 1897/Public Domain
剑桥大学出版社，1897 年/公共领域
Calculus immediately became useful throughout math and physics.
微积分立即在数学和物理学中变得有用。
But it was not formal by modern standards: Leibniz’s and Newton’s papers glossed over some ambiguities.
但以现代标准来看，它并不正式：莱布尼茨和牛顿的论文掩盖了一些含糊之处。
Calculus relies on the notions of infinity and infinitely small quantities (called infinitesimals), but Newton and Leibniz defined these concepts in vague geometric terms; used incorrectly, their formulas could lead to nonsensical calculations, like division by zero.
微积分依赖于无穷大和无限小量（称为无穷小）的概念，但牛顿和莱布尼茨用模糊的几何术语定义了这些概念；如果使用不当，它们的公式可能会导致无意义的计算，例如除以零。
For 150 years, that didn’t cause any problems.
150 年来，这没有造成任何问题。
Scientists simply learned to tell when calculus could be used effectively and when it couldn’t.
科学家们只是学会了判断微积分何时可以有效使用，何时不能有效使用。
But in the early 19th century, mathematicians started to encounter phenomena — infinite sums and strange, jagged curves , for instance — that defied their intuition for what was possible.
但在 19 世纪初，数学家开始遇到一些现象——例如无限和和奇怪的锯齿状曲线——这些现象违背了他们对可能性的直觉。
These encounters came at a time of broader questioning across the arts and sciences.
这些遭遇发生在艺术和科学领域受到更广泛质疑的时期。
Philosophers, painters, writers, and researchers were beginning to probe the limits of what they thought they knew.
哲学家、画家、作家和研究人员开始探索他们自认为所知的极限。
“Intuition is suspect,” said Alma Steingart , a historian at Columbia University.
“直觉是值得怀疑的，”哥伦比亚大学历史学家阿尔玛·斯坦加特说。
“There is this feeling that intuition can lead you the wrong way.”
“有一种感觉，直觉可能会把你引向错误的道路。”
Gottfried Leibniz (left) and Isaac Newton independently invented calculus in the 17th century.
戈特弗里德·莱布尼茨（左）和艾萨克·牛顿在 17 世纪独立发明了微积分。
For years, they feuded over who came up with its central ideas first.
多年来，他们一直在争论谁最先提出其中心思想。
Christoph Bernhard Francke (left); Godfrey Kneller
克里斯托夫·伯恩哈德·弗兰克（左）；戈弗雷·内勒
Renowned mathematicians such as Niels Abel, Augustin-Louis Cauchy, and Karl Weierstrass realized that to make sense of the bizarre sums and curves that were cropping up in their work, they needed to return to their most fundamental definitions.
尼尔斯·阿贝尔、奥古斯丁·路易斯·柯西和卡尔·维尔斯特拉斯等著名数学家意识到，为了理解他们工作中突然出现的奇怪的求和和曲线，他们需要回到最基本的定义。
What is a function?
什么是函数？
What does it mean for a function to be continuous?
函数连续意味着什么？
What really happens when you add up infinitely many objects?
当你将无限多个对象加起来时，到底会发生什么？
They came up with formal definitions that answered these questions.
他们提出了回答这些问题的正式定义。
(Weierstrass, for instance, introduced the precise definition of a limit that students still use today.) Their new definitions enabled mathematicians to study functions in a much deeper and more rigorous way than they ever had before — ultimately spawning the field of analysis.
（例如，韦尔斯特拉斯引入了学生至今仍在使用的极限的精确定义。）他们的新定义使数学家能够以比以前更深入、更严格的方式研究函数，最终催生了分析领域。
Today, analysis is one of the most important areas in math.
今天，分析是数学中最重要的领域之一。
It allows mathematicians to understand everything from fluid flow and electrical currents to the chemical makeup of far-off stars.
它使数学家能够理解从流体流动和电流到遥远恒星的化学组成的一切。
And it is deeply connected to almost every other type of math, including the far older fields of number theory and geometry.
它与几乎所有其他类型的数学都有着深刻的联系，包括更古老的数论和几何领域。
The formalization of calculus also led to the birth of set theory, which established the rules that underlie modern mathematics.
微积分的形式化也导致了集合论的诞生，它确立了现代数学的规则。
Set theory is now an active area of study in its own right.
集合论现在本身就是一个活跃的研究领域。
In the 19th century, Augustin-Louis Cauchy (left) and Karl Weierstrass redefined key concepts in calculus, giving rise to the modern field of analysis.
19 世纪，奥古斯丁-路易斯·柯西（左）和卡尔·韦尔斯特拉斯重新定义了微积分中的关键概念，催生了现代分析领域。
The Smithsonian Libraries (left); Public Domain
史密森尼图书馆（左）；公共领域
Still, some researchers also faced this new wave of formalization with trepidation.
尽管如此，一些研究人员也对这波新的形式化浪潮感到惶恐不安。
“It is not easy to get up any enthusiasm after it has been artificially cooled by the wet blankets of the rigorists,” the physicist Oliver Heaviside wrote in 1899.
物理学家奥利弗·赫维赛德 (Oliver Heaviside) 在 1899 年写道：“在被严格主义者的湿毯子人为冷却之后，再想起来任何热情都不容易。”
As the historian Jesper Lützen has observed , looking back at this time period: “Analysis gained both rigor and generality, but it lost in elegance and simplicity and was estranged from intuition.
正如历史学家耶斯珀·吕岑（Jesper Lützen）回顾这一时期时所观察到的那样：“分析获得了严谨性和普遍性，但失去了优雅和简单性，并疏远了直觉。
Many mathematicians regretted this tendency, but it was hard to escape.”
许多数学家对这种倾向感到遗憾，但又难以逃脱。”
Crucially, rigorists and their detractors found a compromise: Mathematicians continued to use the careful definitions of Cauchy and Weierstrass, but they also developed frameworks that allowed scientists like Heaviside to compute with infinity and infinitesimals more freely.
至关重要的是，严格主义者和他们的批评者找到了一个折衷方案：数学家继续使用柯西和维尔斯特拉斯的仔细定义，但他们也开发了框架，使海维赛这样的科学家能够更自由地计算无穷大和无穷小。
It forces you to think about mathematics in the right way.
它迫使你以正确的方式思考数学。
It forces you to become an artist.
它迫使你成为一名艺术家。
Kevin Buzzard, Imperial College London
凯文·巴扎德，伦敦帝国学院
In other words, formalization wasn’t their only aim.
换句话说，正规化并不是他们的唯一目标。
In fact, an important part of this story is the intent behind the formalization.
事实上，这个故事的一个重要部分是形式化背后的意图。
Its focus was relatively narrow: Weierstrass and his colleagues had been dealing with very particular questions about the behavior of functions, and they turned to formalization in order to deal with those problems.
它的关注点相对狭窄：Weierstrass 和他的同事一直在处理有关函数行为的非常具体的问题，他们转向形式化来解决这些问题。
From there, the development of analysis, set theory, and other formal systems happened organically.
从那时起，分析、集合论和其他形式系统有机地发展起来。
“The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms,” the great mathematician David Hilbert wrote in 1905 .
伟大的数学家戴维·希尔伯特 (David Hilbert) 在 1905 年写道：“科学大厦并不像住宅一样，先打好地基，然后再建造和扩大房间。”
Rather, scientists should first find “comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, [should they] support and fortify them.”
相反，科学家应该首先找到“舒适的空间来漫步，只有随后，当到处出现迹象表明松散的地基无法维持房间的扩张时，[应该]支撑和加固它们。”
“This is not a weakness,” he continued, “but rather the right and healthy path of development.”
“这不是弱点，”他继续说道，“而是正确、健康的发展道路。”
After all, a formalization effort can have very different consequences when it aims to go beyond cementing loose foundations.
毕竟，当形式化努力的目的不仅仅是巩固松散的基础时，它可能会产生截然不同的后果。
It’s possible to take clarity and rigor too far.
清晰度和严谨性可能会太过分。
Consider one of math’s most famous (or infamous, depending on whom you ask) schools of thought, a philosophy put forth by a secret society of mathematicians called Bourbaki.
考虑一下数学中最著名（或臭名昭著，取决于你问的是谁）的思想流派之一，这是由一个名为布尔巴基（Bourbaki）的数学家秘密社团提出的哲学。
In the mid-1930s, mathematics in France had been suffering for decades.
20 世纪 30 年代中期，法国的数学经历了数十年的苦难。
The country had lost a generation of promising students and researchers during World War I; its universities were teaching math in uncoordinated, fragmented ways, using materials that were woefully out of date.
第一次世界大战期间，这个国家失去了一代有前途的学生和研究人员；它的大学以不协调、分散的方式教授数学，使用的材料严重过时。
And so several young mathematicians banded together to form Bourbaki.
于是几位年轻的数学家联合起来成立了布尔巴基。
They began with a modest goal: to bring the French curriculum into the modern era with a new, more cohesive textbook.
他们一开始的目标并不大：用一本新的、更具凝聚力的教科书将法语课程带入现代时代。
Soon, though, their ambitions grew much larger.
但很快，他们的野心就变得更大了。
Today, Bourbaki (whose members remain a secret) has produced over 40 volumes, covering every conceivable area of mathematics.
如今，布尔巴基（其成员仍然是个秘密）已经出版了 40 多本书，涵盖了数学的每一个领域。
The secret math society Bourbaki — founded in the 1930s — valued rigor, abstraction, and logic above all else.
成立于 20 世纪 30 年代的秘密数学社团 Bourbaki 将严谨性、抽象性和逻辑性看得高于一切。
Among its founding members were Henri Cartan (standing, far left), André Weil (standing, second from right) and Szolem Mandelbrojt (seated, right).
其创始成员包括亨利·嘉当（Henri Cartan）（站立，最左边）、安德烈·韦尔（André Weil）（站立，右二）和索尔姆·曼德尔布罗伊特（Szolem Mandelbrojt）（坐，右）。
The real legacy of these books, however, is not their content but their style.
然而，这些书的真正遗产不是它们的内容，而是它们的风格。
Bourbaki prioritized abstraction over all else, eschewing concrete examples and calculations in favor of only the most general statements.
布尔巴基将抽象置于一切之上，避开具体的例子和计算，只支持最笼统的陈述。
They presented each proof as a series of logical deductions, usually without reference to any underlying intuition or rationale.
他们将每个证明呈现为一系列逻辑推论，通常不参考任何潜在的直觉或基本原理。
“It’s a style that is very formal,” said Leo Corry , a historian at Tel Aviv University.
“这是一种非常正式的风格，”特拉维夫大学历史学家利奥·科里 (Leo Corry) 说。
“Very austere.”
“非常严肃。”
Bourbaki’s philosophy quickly spread, influencing mathematics almost everywhere.
布尔巴基的哲学迅速传播，影响了几乎所有地方的数学。
“It had an enormous influence,” said Patrick Massot of Paris-Saclay University.
“它产生了巨大的影响，”巴黎萨克雷大学的帕特里克·马索特说。
“The most successful parts have become so much part of the common mathematical knowledge and style that it’s hard to think of what it was before.”
“最成功的部分已经成为常见数学知识和风格的一部分，以至于很难想象它以前是什么样子。”
The subject became far more airtight, if increasingly abstract and difficult.
这个主题变得更加无懈可击，尽管变得越来越抽象和困难。
“This was not a brilliant decision from a pedagogical point of view,” wrote Maurice Mashaal , the longtime editor of Pour la Science and the author of a book on Bourbaki.
“从教学的角度来看，这不是一个明智的决定，”《为科学而奋斗》的长期编辑、一本关于布尔巴基的书的作者莫里斯·马沙尔 (Maurice Mashaal) 写道。
But the group’s emphasis on abstraction molded research mathematics in ways that are harder to assess.
但该小组对抽象的强调以更难以评估的方式塑造了研究数学。
Some mathematicians revere Bourbaki’s approach.
一些数学家崇敬布尔巴基的方法。
Massot argues that through abstraction and elegance, “you are forced to understand what really makes things work, and what is just noise.” Formalization, in this view, has brought clarity.
马索特认为，通过抽象和优雅，“你被迫了解什么真正让事情发挥作用，什么只是噪音。”从这个角度来看，形式化带来了清晰度。
But Bourbaki’s project had unforeseen consequences as well.
但布尔巴基的项目也产生了不可预见的后果。
Their vocabulary and style weren’t a natural fit for every type of mathematics.
他们的词汇和风格并不适合每种类型的数学。
For example, the fields of combinatorics (often described as the science of counting) and graph theory (the science of networks) tend to be highly concrete and visual.
例如，组合学（通常被描述为计数科学）和图论（网络科学）领域往往是高度具体和可视化的。
Perhaps, then, it’s no surprise that for decades, they were sidelined at most prestigious institutions in the United States and parts of Europe; they were only able to thrive in places where Bourbaki’s influence was limited, like Hungary.
那么，几十年来，他们在美国和欧洲部分地区最负盛名的机构中被边缘化，也许也就不足为奇了。他们只能在布尔巴基影响力有限的地方蓬勃发展，比如匈牙利。
“Graph theory [was] the slum of topology,” said Béla Bollobás of the University of Cambridge.
“图论是拓扑学的贫民窟，”剑桥大学的贝拉·博洛巴斯 (Béla Bollobás) 说。
“That atmosphere changed only recently.
“这种气氛直到最近才发生变化。
Very recently.”
就在最近。”
Aravind Asok worries that overemphasizing formalization might homogenize mathematics.
阿拉文德·阿索克（Aravind Asok）担心，过分强调形式化可能会使数学同质化。
Even in fields that did flourish under the Bourbaki framework — algebra, topology, analysis — some mathematicians worry that Bourbaki was too successful.
即使在布尔巴基框架下确实蓬勃发展的领域——代数、拓扑、分析——一些数学家也担心布尔巴基过于成功。
According to them, the ways in which proofs are written and theories constructed have become overly homogeneous.
根据他们的说法，证明的书写方式和理论的构建方式已经变得过于同质化。
“There is a sense that it decreased the cultural diversity of mathematics,” said Aravind Asok of the University of Southern California.
“有一种感觉，它降低了数学的文化多样性，”南加州大学的阿拉文德·阿索克（Aravind Asok）说。
Before Bourbaki, for instance, there were many versions of algebraic geometry.
例如，在布尔巴基之前，代数几何有许多版本。
In France, for example, methods were grounded in analysis, while in Italy, a geometric style reigned.
例如，在法国，方法以分析为基础，而在意大利，几何风格占主导地位。
The Italian school, which lacked rigor and involved many errors, quickly faded as Bourbaki’s influence spread.
意大利学派缺乏严谨性，错误较多，随着布尔巴基影响力的扩大，意大利学派很快就衰落了。
Certainly, algebraic geometry became more reliable.
当然，代数几何变得更加可靠。
But by cutting off other paths of possible progress, Bourbaki introduced a new problem.
但通过切断其他可能取得进展的途径，布尔巴基引入了一个新问题。
“I don’t want any mathematics where one mode dominates,” Asok said.
“我不想要任何一种模式占主导地位的数学，”阿索克说。
“There is a cultural history to mathematics that deserves to be preserved.”
“数学有一段值得保存的文化历史。”
Despite the best efforts of Bourbaki, Cauchy, and Weierstrass, truly formal proofs have always been objects of theory, not practice.
尽管布尔巴基、柯西和维尔斯特拉斯做出了最大的努力，真正的形式证明始终是理论的对象，而不是实践的对象。
Some mathematicians now hope that computers can change that.
一些数学家现在希望计算机能够改变这一现状。
Since the 1960s, researchers have been developing computer programs called proof assistants.
自 20 世纪 60 年代以来，研究人员一直在开发称为证明助手的计算机程序。
Using a proof assistant, a mathematician will write every line of a proof (including every definition) in a language that the computer can understand, and then the proof assistant will check the logic.
使用证明助手，数学家将用计算机可以理解的语言编写证明的每一行（包括每个定义），然后证明助手将检查逻辑。
If even a single step doesn’t follow from a preceding one — if you haven’t rigorously shown every tiny detail, like the fact that 1 + 1 equals 2 — then the program will not consider the proof correct.
如果甚至有一个步骤不遵循前一个步骤——如果你没有严格地显示每一个微小的细节，比如 1 + 1 等于 2 的事实——那么程序将不会认为证明是正确的。
Currently, mathematicians are hoping to formalize all of mathematics using a proof assistant called Lean.
目前，数学家希望使用名为 Lean 的证明助手来形式化所有数学。
They’ve created a library of more than 120,000 definitions so far and have verified a quarter of a million theorems.
到目前为止，他们已经创建了一个包含超过 120,000 个定义的库，并验证了 25 万个定理。
Several mathematicians maintain this database, keeping it up to date and vetting new contributions.
几位数学家维护这个数据库，使其保持最新状态并审查新的贡献。
(A handful of them do this work full time.) They’ve received over $10 million in funding, mostly from the billionaire financier Alex Gerko.
（他们中的少数人全职从事这项工作。）他们已经收到了超过 1000 万美元的资金，其中大部分来自亿万富翁金融家 Alex Gerko。
Samuel Velasco/ Quanta Magazine
塞缪尔·贝拉斯科/广达杂志
Lean is “revolutionary,” said Alex Kontorovich of Rutgers University, in part because it breaks up a single proof into many pieces that can each be handled separately, verified independently, and reused in other proofs.
罗格斯大学的 Alex Kontorovich 表示，精益是“革命性的”，部分原因是它将单个证明分解为多个部分，每个部分都可以单独处理、独立验证，并在其他证明中重复使用。
“Imagine if every time you wanted to build a spaceship, every single engineer has to understand every single component — from mining the iron ore, to smelting it, designing it.
“想象一下，如果每次你想建造一艘宇宙飞船，每一位工程师都必须了解每一个组件——从开采铁矿石到冶炼、设计。
Now with these formal systems, for the first time, you can do mathematics and just buy somebody’s part off the shelf.”
现在有了这些正式的系统，你第一次可以做数学计算，然后直接购买别人的现成零件。”
I don’t want any mathematics where one mode dominates.
我不想要任何一种模式占主导地位的数学。
There is a cultural history to mathematics that deserves to be preserved.
数学有一段值得保存的文化历史。
Aravind Asok, University of Southern California
阿拉文德·阿索克，南加州大学
But it usually takes months to write and vet definitions and proofs in Lean.
但通常需要几个月的时间来编写和审查精益的定义和证明。
(In some cases, formalizing a proof takes only a few weeks; in others, more than a year.) Some mathematicians therefore worry that their time and resources would be better spent on other pursuits.
（在某些情况下，形式化证明只需要几周的时间；在其他情况下，需要一年多的时间。）因此，一些数学家担心他们的时间和资源最好花在其他追求上。
They argue that while verifying proofs is important, checking them by hand has worked well enough.
他们认为，虽然验证证据很重要，但手工检查已经足够好了。
Even though “there are many, many errors in the literature,” Asok said, “my experience is that mathematics is remarkably robust.” There’s little danger, in other words, that the whole edifice of math could come crashing down.
尽管“文献中有很多很多错误，”阿索克说，“但我的经验是数学非常强大。”换句话说，整个数学大厦倒塌的危险很小。
That said, using Lean has also led to new math.
也就是说，使用精益也带来了新的数学。
In 2019, the mathematician Peter Scholze wrote a proof by hand of a theorem that was poised to play a key role in a new mathematical theory he was developing.
2019 年，数学家 Peter Scholze 手写了一个定理的证明，该定理有望在他正在开发的新数学理论中发挥关键作用。
But the proof was extremely complicated, and he struggled to determine if it was correct.
但证明极其复杂，他很难确定它是否正确。
So in late 2020, a team of mathematicians led by Johan Commelin and Adam Topaz set out to formalize the proof in Lean .
因此，在 2020 年底，由 Johan Commelin 和 Adam Topaz 领导的数学家团队开始用 Lean 形式化证明。
Several months later, they confirmed that it was correct, giving mathematicians greater confidence in Scholze’s new theory.
几个月后，他们证实这是正确的，让数学家对舒尔茨的新理论更有信心。
And in the process, they found a cleaner proof and refined some of Scholze’s original ideas.
在此过程中，他们找到了更清晰的证据，并完善了 Scholze 的一些原始想法。
“It forces you to think about mathematics in the right way,” said Kevin Buzzard , a mathematician at Imperial College London and one of Lean’s biggest proponents.
“它迫使你以正确的方式思考数学，”伦敦帝国理工学院的数学家、精益的最大支持者之一凯文·巴扎德 (Kevin Buzzard) 说。
“It forces you to become an artist.” (Buzzard has spent the past year working to use Lean to formalize a proof of Fermat’s Last Theorem , an important result whose proof is famously long and complicated.)
“它迫使你成为一名艺术家。” （Buzzard 在过去的一年里一直致力于使用 Lean 来形式化费马大定理的证明，这是一个重要的结果，其证明又长又复杂。）
Kevin Buzzard is currently using Lean to formalize the proof of Fermat’s Last Theorem, one of the most famous results in math.
Kevin Buzzard 目前正在使用 Lean 来形式化费马大定理的证明，这是数学界最著名的结果之一。
“I want this argument to be a thing of beauty,” he said.
“我希望这场争论是一件美丽的事情，”他说。
“I want it to slip down the throat.”
“我想让它滑进喉咙。”
Moreover, taking the time to develop Lean now might be a good investment in a future where artificial intelligence is bound to play a bigger role in mathematics.
此外，现在花时间发展精益可能是对未来人工智能必将在数学中发挥更大作用的良好投资。
As mathematicians try to use AI to write informal proofs, it will be more important to verify their accuracy using programs like Lean.
当数学家尝试使用人工智能来编写非正式证明时，使用 Lean 等程序来验证其准确性将变得更加重要。
(Plus, AI is already helping to write Lean proofs more efficiently.)
（此外，人工智能已经在帮助更有效地编写精益证明。）
Still, the widespread adoption of Lean comes with risks.
尽管如此，精益的广泛采用也伴随着风险。
Could Lean, like Bourbaki, subtly shift the kinds of questions that mathematicians find interesting?
精益能否像布尔巴基一样巧妙地改变数学家感兴趣的问题类型？
Within Lean, there is little room for conceptual, methodological, or ideological diversity.
在精益内部，概念、方法论或意识形态多样性几乎没有空间。
For one thing, every new proof in Lean can only use formal definitions and theorems that have already been verified and stored in its library.
一方面，精益中的每个新证明只能使用已经验证并存储在其库中的形式定义和定理。
This means that these definitions and proofs need to seamlessly fit together.
这意味着这些定义和证明需要无缝地结合在一起。
It also means that updating or changing a definition creates a domino effect: Proofs written with the old definition may not work with the new one.
这也意味着更新或更改定义会产生多米诺骨牌效应：用旧定义编写的证明可能不适用于新定义。
That could be a problem.
这可能是个问题。
Mathematics is “not a fixed and finite and formalized enterprise,” said Stephanie Dick of Simon Fraser University.
西蒙弗雷泽大学的斯蒂芬妮迪克说，数学“不是一个固定的、有限的和形式化的事业”。
“It’s a moving, shifting beast, and the terms of it are changing all the time.”
“它是一头移动、变化的野兽，它的术语一直在变化。”
Stephanie Dick is wary of how new technologies, such as proof assistants, might subtly influence what questions mathematicians ask in their research.
斯蒂芬妮·迪克（Stephanie Dick）对证明助手等新技术可能会微妙地影响数学家在研究中提出的问题持谨慎态度。
Eric Sucar/University of Pennsylvania
埃里克·苏卡尔/宾夕法尼亚大学
Because of this, previous attempts to digitize mathematical results (such as a 1994 project called the QED Manifesto ) quickly deteriorated into arguments about which definitions and frameworks to use.
因此，之前对数学结果进行数字化的尝试（例如 1994 年名为 QED 宣言的项目）很快就陷入了关于使用哪些定义和框架的争论。
“Everybody loves this idea in principle, but in practice, it’s a nightmare,” Dick said.
“原则上每个人都喜欢这个想法，但实际上，这是一场噩梦，”迪克说。
“They disagree about what programming language to write in.
“他们对于用什么编程语言编写存在分歧。
…
……
There’s fundamental disagreement about whether this kind of project could be possible at all.”
对于这种项目是否可行存在根本分歧。”
To avoid these disagreements, a dedicated group of Lean users is responsible for determining which definitions should go into Lean’s library and how those definitions should be coded — much as Wikipedia does it, according to Simon DeDeo of Carnegie Mellon University.
卡内基梅隆大学的西蒙·德迪奥 (Simon DeDeo) 表示，为了避免这些分歧，专门的精益用户小组负责确定哪些定义应进入精益库以及这些定义应如何编码——就像维基百科所做的那样。
“It is really a community trying to do the best thing for the community,” said Emily Riehl , a mathematician at Johns Hopkins University.
“这确实是一个试图为社区做最好事情的社区，”约翰·霍普金斯大学数学家艾米丽·里尔 (Emily Riehl) 说。
So far, it’s worked, and the process has been largely democratic.
到目前为止，它是有效的，而且这个过程基本上是民主的。
But, she said, “what’s bad about it is, there is one decision that will be made in the end, [when] in many cases, there’s not one correct decision.
但是，她说，“不好的是，最终会做出一个决定，但在很多情况下，没有一个正确的决定。
Some people will be happy, and other people will be unhappy.”
有些人会快乐，有些人会不快乐。”
Just as Bourbaki’s approach was optimal for certain subjects and not others, the Lean language and the choice of definitions that go into its library are better suited to some areas of math than others.
正如布尔巴基的方法对于某些学科而不是其他学科来说是最佳的一样，精益语言和进入其库的定义选择比其他领域更适合数学的某些领域。
They’re a good fit for number theory and algebraic geometry, for instance.
例如，它们非常适合数论和代数几何。
Graph theory and category theory, not so much.
图论和范畴论，没那么多。
This is just one way Lean could make math more homogeneous.
这只是精益使数学更加均匀的一种方式。
Dick points out that past technologies — even choices of which notation to use — have subtly changed how mathematicians approach their work.
迪克指出，过去的技术——甚至是选择使用哪种符号——已经巧妙地改变了数学家处理工作的方式。
Lean might encourage them to focus on developing concepts that can be formalized more easily, even if they don’t realize it.
精益可能会鼓励他们专注于开发更容易形式化的概念，即使他们没有意识到这一点。
“I’ve now found so many cases where something like that is happening,” she said — where “it actually shifts the focus and the intuition and the understanding away from the mathematical problem domain toward the behavior of this system.”
“我现在发现很多类似的情况正在发生，”她说，“它实际上将焦点、直觉和理解从数学问题领域转移到了这个系统的行为上。”
There is certainly a lot to be excited about when it comes to Lean.
说到精益，肯定有很多令人兴奋的事情。
But Riehl, for her part, thinks that mathematicians should try to use more than one kind of proof assistant, rather than relying solely on Lean.
但里尔认为，数学家应该尝试使用不止一种证明助手，而不是仅仅依赖精益。
She’s not sure how practical that will be, however, given how much effort formalization requires.
然而，考虑到正规化需要付出多少努力，她不确定这有多实用。
For centuries now, mathematicians have sought greater rigor by focusing on whether they can verify every line of a proof.
几个世纪以来，数学家一直在关注是否能够验证证明的每一行，以寻求更严格的结果。
But that wasn’t always the case.
但情况并非总是如此。
To Descartes in the 1600s, rigor meant being able to hold an idea in your head and intuitively understand why it’s true.
对于 1600 年代的笛卡尔来说，严谨意味着能够在头脑中牢记一个想法并直观地理解为什么它是正确的。
As a result, according to Akshay Venkatesh of the Institute for Advanced Study, older proofs used to “have a kind of grittiness.”
因此，根据高级研究所的阿克谢·文卡特什 (Akshay Venkatesh) 的说法，旧的证明过去“有一种坚韧”。
“It doesn’t make for a formally elegant argument,” he said.
“这并不构成正式优雅的论证，”他说。
But it’s “something a human being can easily understand.” Modern proofs are more elegant, sure.
但这是“人类很容易理解的事情”。当然，现代的证明更加优雅。
But they’re also more slippery, harder for the mind to get a grip on.
但它们也更狡猾，更难被大脑抓住。
(Interestingly, Buzzard used similar language while discussing his hopes for how Lean would influence proof-writing.
（有趣的是，巴扎德在讨论他对精益如何影响校对写作的希望时使用了类似的语言。
“I want this argument to be a thing of beauty,” he said.
“我希望这场争论是一件美丽的事情，”他说。
“I want it to slip down the throat.”)
“我想让它滑进喉咙。”）
This trend reflects a notion that’s now taken for granted: that proof should lie at the heart of math.
这种趋势反映了一个现在被认为是理所当然的观念：证明应该是数学的核心。
“There’s no question that the concept of proof is a fundamental part of mathematics,” Venkatesh said.
“毫无疑问，证明的概念是数学的基本组成部分，”文卡特什说。
But “where I think there should be a question is whether it should be the defining feature of mathematics.”
但“我认为应该有一个问题是它是否应该成为数学的定义特征。”
Formalization in Lean would continue this trend of prioritizing proof.
精益的形式化将延续这种优先证明的趋势。
But it’s not the only future that mathematicians are imagining.
但这并不是数学家想象的唯一未来。
Researchers are encouraged to think “that the only way forward is to have papers be Lean-formalized,” Asok said.
阿索克说，研究人员被鼓励认为“唯一的出路就是让论文精益形式化”。
“I would argue another way is that we just take a step back and stop writing so much.
“我认为另一种方式是我们退后一步，停止写那么多。
But that’s antithetical to the incentives that are in place.”
但这与现有的激励措施背道而驰。”
It’s impossible to predict all the possible ways that formalization in Lean could impact mathematics.
不可能预测精益形式化对数学产生影响的所有可能方式。
What’s clear from history is that math has a tendency to self-correct — and that the future of this next wave of formalization will be one we haven’t imagined yet.
从历史中可以清楚地看出，数学具有自我纠正的倾向，而下一波形式化的未来将是我们尚未想象到的。
Editor’s Note: Alex Kontorovich is a member of Quanta Magazine ’s advisory board.
编者注：Alex Kontorovich 是《Quanta》杂志顾问委员会的成员。
Correction: March 27, 2026 An earlier version of this story referred to Maurice Mashaal as a mathematician.
更正：2026 年 3 月 27 日这个故事的早期版本将莫里斯·马沙尔 (Maurice Mashaal) 称为数学家。
He is in fact a journalist.
他实际上是一名记者。
Also, Béla Bollobás’s affiliation has been updated to reflect that he is primarily affiliated with the University of Cambridge.
此外，Béla Bollobás 的隶属关系已更新，以反映他主要隶属于剑桥大学。
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