Can the Most Abstract Math Make the World a Better Place?
“I ’ve spent a long time exploring the crystalline beauty of traditional mathematics, but now I’m feeling an urge to study something slightly more earthy,” John Baez wrote on his blog in 2011.
“我花了很长时间探索传统数学的水晶之美，但现在我有一种冲动去研究一些更朴实的东西，”约翰·贝兹 (John Baez) 2011 年在他的博客中写道。
An influential mathematical physicist who splits his time between the University of California, Riverside and the University of Edinburgh, Baez had grown increasingly concerned about the state of the planet, and he thought mathematicians could do something about it.
贝兹是一位颇具影响力的数学物理学家，他的时间分布在加州大学河滨分校和爱丁堡大学，他越来越关心地球的状况，他认为数学家可以为此做点什么。
Baez called for the development of new mathematics — he called it “green” math — to better capture the workings of Earth’s biosphere and climate.
贝兹呼吁发展新的数学——他称之为“绿色”数学——以更好地捕捉地球生物圈和气候的运作情况。
For his part, he sought to apply category theory, a highly abstract branch of math in which he is an expert, to modeling the natural world.
就他而言，他寻求应用范畴论（他是该领域的专家）的高度抽象的数学分支来建模自然世界。
It sounds like a pipe dream.
这听起来像是一个白日梦。
Math works well at describing simple, isolated systems, but as we go from atoms to organisms to ecosystems, concise mathematical models typically become less effective.
数学很适合描述简单、孤立的系统，但当我们从原子到有机体再到生态系统时，简洁的数学模型通常会变得不太有效。
The systems are just too complex.
这些系统太复杂了。
I n philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted.
在哲学中，“感受性”指的是我们体验的主观品质：爱丽丝看到蓝色或鲍勃感到高兴是什么感觉。
Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it.
正如已故哲学家丹尼尔·丹尼特所说，感受性是“事物在我们看来的方式”。
In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
在这些文章中，我们的专栏作家追随他们的好奇心，探索重要但不一定可以回答的科学问题。
But in the years since Baez’s post, more than 100 mathematicians have joined him as “applied category theorists” attempting to model a variety of real-world systems in a new way.
但自 Baez 上任以来的几年里，已有 100 多名数学家加入他的行列，成为“应用范畴理论家”，试图以新的方式对各种现实世界系统进行建模。
Applied category theory now has an annual conference, an academic journal, and an institute, as well as a research program funded by the U.K. government.
应用范畴论现在有一个年度会议、一份学术期刊、一个研究所以及一个由英国政府资助的研究项目。
Skepticism abounds, however.
然而，怀疑论比比皆是。
“When I say we’re underdogs and nobody likes us, it’s not completely true, but it’s a bit true,” one applied category theorist, Matteo Capucci , told me.
“当我说我们处于劣势，没有人喜欢我们时，这并不完全正确，但确实有点正确，”一位应用范畴理论家马泰奥·卡普奇 (Matteo Capucci) 告诉我。
I set out to learn what this burgeoning research area is about.
我开始了解这个新兴研究领域的内容。
How could one of the seemingly most rarefied realms of pure math help demystify a system as complex as the biosphere?
看似最稀有的纯数学领域之一如何帮助揭开像生物圈这样复杂的系统的神秘面纱？
Is it a significant improvement on other approaches to modeling?
与其他建模方法相比，它是否有显着改进？
Can mathematics really be green?
数学真的可以绿色吗？
It didn’t seem promising.
看起来前景并不乐观。
To my surprise, I’ve learned that applied category theory has had some wins lately.
令我惊讶的是，我了解到应用范畴论最近取得了一些胜利。
The applications are not yet as green as Baez had hoped, but the approach is showing potential in important areas, including epidemiology and artificial intelligence safety.
这些应用程序尚未像贝兹所希望的那样绿色，但该方法在流行病学和人工智能安全等重要领域显示出潜力。
It seems plausible that the most abstract idealizations can help make greater sense of the messiest realities.
最抽象的理想化可以帮助我们更好地理解最混乱的现实，这似乎是有道理的。
Category theory originated in 1945 as an effort to formalize relationships between mathematical objects, and it soon grew into a powerful and productive branch of math.
范畴论起源于 1945 年，旨在形式化数学对象之间的关系，很快就发展成为一个强大且富有成效的数学分支。
What do we mean by mathematical objects?
数学对象是什么意思？
Numbers, functions, and sets are examples.
数字、函数和集合都是示例。
To a category theorist, what defines an object is its relationships to others.
对于范畴论学家来说，定义一个对象是它与其他对象的关系。
What is a black king in chess?
国际象棋中的黑王是什么？
“You can say it’s a bit of wood carved into a certain shape and painted black, but that’s not important; it could be a saltshaker,” said Tom Leinster , a mathematician at the University of Edinburgh.
爱丁堡大学数学家汤姆·伦斯特 (Tom Leinster) 表示：“你可以说它是一块雕刻成特定形状并涂成黑色的木头，但这并不重要；它可能是一个盐瓶。”
Rather, the black king is defined by how it moves on a chessboard and how it can capture opposing pieces or be checked by them.
相反，黑王的定义是它如何在棋盘上移动以及如何吃掉对方棋子或被对方棋子检查。
Jeremy Howard via flickr, modified by Quanta
Jeremy Howard 来自 flickr，由 Quanta 修改
A category is a collection of objects and these relationships, or morphisms.
范畴是对象和这些关系或态射的集合。
Let’s consider that chess set as a category.
让我们将国际象棋视为一个类别。
To do so, you might depict it as a diagram featuring little boxes for each object — legal chess positions — and then connect the boxes with arrows to represent the morphisms, or possible moves.
为此，您可以将其描述为一个图表，其中每个对象都有小方框（合法的国际象棋位置），然后用箭头连接这些方框以表示态射或可能的移动。
Category theorists study how to map, overlap, or connect various categories.
类别理论家研究如何映射、重叠或连接各种类别。
We’re all intuitively aware of categories.
我们都凭直觉知道类别。
We know, for example, that 5 feet and $5 are not the same mathematically.
例如，我们知道 5 英尺和 5 美元在数学上并不相同。
You can multiply 5 feet by 3 feet to get 15 square feet.
您可以将 5 英尺乘以 3 英尺得到 15 平方英尺。
But you can’t multiply $5 and $3 — there’s no such thing as square dollars.
但你不能将 5 美元和 3 美元相乘——不存在平方美元这样的东西。
You can add $5 and $3, or you can multiply $5 by 3 (the number, not dollars).
您可以将 5 美元和 3 美元相加，也可以将 5 美元乘以 3（数字，而不是美元）。
But $5 times $3 is meaningless.
但 5 美元乘以 3 美元毫无意义。
Despite knowing nothing of vector spaces or morphisms, we somehow manage to avoid making embarrassing category errors at the checkout counter.
尽管对向量空间或态射一无所知，我们还是设法避免在结账柜台出现令人尴尬的类别错误。
But when concepts are more varied and complex than distances and dollars, we run into problems.
但当概念比距离和金钱更加多样化和复杂时，我们就会遇到问题。
“It shows up all the time in modeling, for example epidemiological modeling,” Baez told me.
“它一直出现在建模中，例如流行病学建模中，”贝兹告诉我。
“If you’re writing a model in conventional software and you type ‘35’ into your program, that doesn’t tell you whether it’s 35 dollars or 35 people or 35 doses of a drug.
“如果你在传统软件中编写模型，并在程序中输入‘35’，这并不能告诉你它是 35 美元、35 人还是 35 剂药物。
And so you’re conflating those all as just numbers, and that makes it easier to make mistakes.”
所以你把这些都当作数字混为一谈，这样就更容易犯错误。”
Applied category theory provides a framework for modeling real-world systems in terms of objects and morphisms.
应用范畴论提供了一个根据对象和态射对现实世界系统进行建模的框架。
“Categories are ways of organizing logical structures,” said Brendan Fong, the co-founder and CEO of the Topos Institute in Berkeley, which is devoted to applications of category theory.
“范畴是组织逻辑结构的方式，”伯克利 Topos 研究所的联合创始人兼首席执行官 Brendan Fong 说，该研究所致力于范畴论的应用。
The physicist Bob Coecke applied it to quantum mechanics in the 2000s, which has since been extended to reasoning about quantum computation.
物理学家 Bob Coecke 在 2000 年代将其应用于量子力学，此后又扩展到量子计算的推理。
A few years later, Baez started mulling over the categorization of the biosphere, while the mathematician David Spivak, who co-founded the Topos Institute, independently pioneered applied category theory by thinking about databases.
几年后，贝兹开始思考生物圈的分类，而拓扑斯研究所联合创始人的数学家大卫·斯皮瓦克则通过思考数据库独立开创了应用范畴论。
“David has a real imperative to formalize and make legible the world,” Fong said.
“大卫确实有必要将这个世界正规化并变得清晰，”方说。
“The thing he hates most in the world is miscommunication.”
“他在这个世界上最讨厌的就是沟通不畅。”
Xenja Santarelli, modified by Quanta
Xenja Santarelli，由 Quanta 修改
In a 2022 lecture I watched online, Spivak envisioned how applied category theory might work in practice.
在我在线观看的 2022 年讲座中，斯皮瓦克设想了应用范畴论如何在实践中发挥作用。
An accountant tells an applied category theorist about the objects in their database, such as employees and dollar amounts.
会计师向应用类别理论家讲述其数据库中的对象，例如员工和金额。
The category theorist then develops a formal model of the system — a category with a rigorous logical structure — which can then be connected to other categories, corresponding to other databases and spreadsheets, to model the whole company.
然后，类别理论家开发系统的正式模型 - 具有严格逻辑结构的类别 - 然后可以将其连接到与其他数据库和电子表格相对应的其他类别，以对整个公司进行建模。
In this way, applied category theory is a lingua franca for talking about the heterogeneous parts of some giant system.
这样，应用范畴论就成为讨论某些巨型系统的异质部分的通用语言。
Climate modeling — one of Baez’s initial candidates for green mathematics — attempts to simulate the archetypal giant system: Earth itself.
气候建模——贝兹最初的绿色数学候选之一——试图模拟典型的巨型系统：地球本身。
Experts in different parts of the Earth system must assemble their knowledge and streams of data in a logical way to understand the whole.
地球系统不同部分的专家必须以逻辑方式整合他们的知识和数据流，以理解整体。
But Baez and others told me that applied category theorists don’t have any purchase in climate science, partly because climate models are already sophisticated enough to function, despite a lack of mathematical rigor in how they’re patched together.
但贝兹和其他人告诉我，应用范畴理论家对气候科学没有任何购买力，部分原因是气候模型已经足够复杂，可以发挥作用，尽管它们的拼凑方式缺乏数学严谨性。
That rigor could make models stronger, more flexible, and better able to integrate new information, the mathematicians argue, but starting over takes both convincing and effort.
数学家认为，这种严谨性可以使模型更强大、更灵活，并且能够更好地整合新信息，但重新开始既需要令人信服的努力，又需要付出努力。
“It’s one of the challenges we always face in applied category theory,” said Amar Hadzihasanovic of Tallinn University of Technology in Estonia.
“这是我们在应用范畴论中始终面临的挑战之一，”爱沙尼亚塔林理工大学的 Amar Hadzihasanovic 说。
“We can go to people and tell them, ‘Your model would be better if you would assemble it according to these first principles.’ And they tell you, ‘OK, well, how long is it going to take?’ It’s a big investment before you can reap the benefits.”
“我们可以去找人们并告诉他们，‘如果你按照这些首要原则来组装你的模型会更好。’他们会告诉你，‘好吧，需要多长时间？’在你获得收益之前，这是一项巨大的投资。”
There’s little math can do to address the inadequate political response to the climate crisis, but applied category theory is further along in other areas of public concern.
几乎没有数学可以解决对气候危机的政治反应不足的问题，但应用类别理论在公众关注的其他领域取得了进一步的进展。
For example, Baez has been collaborating with Topos and a Canadian computer scientist, Nate Osgood , who specializes in epidemiological modeling of disease outbreaks.
例如，Baez 一直在与 Topos 和加拿大计算机科学家 Nate Osgood 合作，后者专门研究疾病爆发的流行病学模型。
While working on Canada’s pandemic response at the University of Saskatchewan, Osgood was frustrated that existing modeling software didn’t allow experts to combine knowledge from different fields.
在萨斯喀彻温大学研究加拿大的流行病应对工作时，奥斯古德对现有的建模软件不允许专家结合不同领域的知识感到沮丧。
Robert Koorenny, modified by Quanta
罗伯特·库伦尼 (Robert Koorenny)，由广达修改
To predict how an outbreak will progress, epidemiologists often use stock-and-flow diagrams: illustrations featuring stocks of people (susceptible, infected, recovered, dead) and arrows showing flows between them based on factors such as exposure or virulence.
为了预测疫情将如何发展，流行病学家经常使用存量流量图：插图以人群（易感、感染、康复、死亡）为特色，箭头根据暴露或毒力等因素显示人群之间的流动。
Stocks and flows are just objects and morphisms of a category.
存量和流量只是一个范畴的对象和态射。
The arrangement of boxes and arrows in the diagrams translates into equations describing the system’s evolution.
图中方框和箭头的排列转化为描述系统演化的方程。
Over the last few years, Osgood, Baez, and their team have developed a software package called StockFlow that formalizes this kind of modeling.
在过去的几年里，Osgood、Baez 和他们的团队开发了一个名为 StockFlow 的软件包，该软件包将这种建模形式化。
Specialists can model different aspects of an outbreak, such as how health disparities affect the infection rates of susceptible people, and these categories can be composed into larger ones.
专家可以对疫情爆发的不同方面进行建模，例如健康差异如何影响易感人群的感染率，并且这些类别可以组成更大的类别。
“Category theory is able to handle those fancier forms of composition,” Baez said.
“范畴论能够处理那些更奇特的组合形式，”贝兹说。
StockFlow has yet to spread among epidemiologists, but Osgood teaches it to his students in hopes of inoculating the next generation of modelers.
StockFlow 尚未在流行病学家中传播，但奥斯古德将其教授给他的学生，希望为下一代建模师提供帮助。
“This is genuinely something that could be used,” said Leinster.
“这确实是可以使用的东西，”伦斯特说。
“It’s serious stuff.”
“这是严肃的事情。”
Meanwhile, Hadzihasanovic and Capucci are both part of Safeguarded AI , a project funded by ARIA, a U.K. government–funded advanced research agency, that’s applying category theory to the problem of AI safety.
与此同时，Hadzihasanovic 和 Capucci 都是 Safeguarded AI 项目的一部分，该项目由英国政府资助的高级研究机构 ARIA 资助，该项目将范畴论应用于人工智能安全问题。
How, the program asks, can unpredictable and error-prone AI systems be trusted to operate essential real-world systems, such as nuclear plants or power grids?
该计划提出，如何才能信任不可预测且容易出错的人工智能系统来运行核电站或电网等重要的现实系统？
I can see the need here, and the team’s answer is clever: Build formal models of complex systems for the AI to practice on.
我可以看到这里的需求，团队的答案很聪明：构建复杂系统的正式模型，供人工智能练习。
These models must have the same logical structure as the real system, correctly representing the morphisms between many different types of objects.
这些模型必须具有与真实系统相同的逻辑结构，正确地表示许多不同类型对象之间的态射。
“Category theory gives you a modular and compositional way of doing this,” Capucci said.
“范畴论为你提供了一种模块化和组合的方式来做到这一点，”卡普奇说。
“We are developing fundamental technology that we can deploy in so many situations.”
“我们正在开发可以在多种情况下部署的基础技术。”
There’s a sense among applied category theorists that their approach will pay off in the long run, as systems grow ever more complex and interconnected, and as AI gets more involved.
应用范畴理论家有一种感觉，随着系统变得越来越复杂和相互关联，以及人工智能越来越多地参与，他们的方法从长远来看将会得到回报。
Winging it won’t cut it.
挥舞它不会削减它。
“This is going to be, eventually, very important work,” Hadzihasanovic said.
“这最终将成为非常重要的工作，”哈齐哈萨诺维奇说。
Many practitioners got into the field because they share Baez’s environmental ethos and hope to take on greener problems in time.
许多从业者进入这个领域是因为他们认同贝兹的环保理念，并希望及时解决更环保的问题。
Baez still has high hopes.
贝兹仍然抱有很大的希望。
A cousin of Joan Baez, the folk singer and activist, he was heavily influenced by his uncle (her father), a physicist who was also a socially active Quaker.
他是民谣歌手和活动家琼·贝兹 (Joan Baez) 的表弟，深受叔叔（她的父亲）的影响，叔叔是一位物理学家，也是一位社会活跃的贵格会教徒。
It’s “infused in me,” he said, to help the world and “not just enjoy myself.”
他说，它“融入了我的内心”，帮助世界，“而不仅仅是让自己快乐”。
I asked what it is about the biosphere that he thinks category theory can help us understand.
我问他认为范畴论可以帮助我们理解生物圈的哪些方面。
In his view, we improperly categorize biological systems.
在他看来，我们对生物系统的分类不正确。
We mistake them for machines, objects that perform specific tasks by taking in matter and energy and producing desired outputs and waste.
我们将它们误认为是机器，是通过吸收物质和能量并产生所需输出和废物来执行特定任务的物体。
“We focus on the part we care about and ignore the waste and where is the energy coming from,” Baez said.
“我们专注于我们关心的部分，而忽略了浪费以及能源来自哪里，”贝兹说。
“Our whole technology and indeed our whole mathematics is based on that attitude.”
“我们的整个技术，甚至我们的整个数学都是基于这种态度。”
Living systems are a different category, though.
不过，生命系统是一个不同的类别。
They’re not built to perform tasks.
它们不是为执行任务而构建的。
Evolution has made life “incredibly subtle and complicated in ways we can’t fully fathom,” he said.
他说，进化使生命“变得极其微妙和复杂，我们无法完全理解”。
Genes aren’t discrete parts of a machine with their own purposes, for instance; they all have numerous roles and impacts.
例如，基因并不是一台机器的离散部分，它们有自己的用途。它们都有许多作用和影响。
In an ecosystem, there’s no waste; one creature’s poop is another’s feast.
在生态系统中，没有浪费；一种生物的粪便是另一种生物的盛宴。
“I don’t think we have the math to understand such systems yet,” said Baez, who thinks modeling these systems will involve new categories with previously unstudied logical structures.
贝兹说：“我认为我们还没有数学知识来理解此类系统。”他认为对这些系统进行建模将涉及具有先前未研究过的逻辑结构的新类别。
“That’s the kind of mathematics I would like to develop, because I have this hope that it will help us be kinder to the world if we understand the world a bit better and not think of the natural world as raw materials for our machines to take advantage of to do what we want to do.
“这就是我想要发展的数学，因为我希望，如果我们更好地了解世界，而不是将自然世界视为我们的机器利用做我们想做的事情的原材料，它将帮助我们对世界更加友善。
That attitude that we have right now is running into a wall.
我们现在的态度正在碰壁。
That attitude winds up destroying the whole planet.”
这种态度最终会摧毁整个地球。”
Indeed, we might value nonhumans, ecosystems, and the climate more by conceiving of them and ourselves as objects in a shared category.
事实上，通过将非人类、生态系统和气候与我们自己视为共享类别中的物体，我们可能会更加重视它们。
Like these mathematicians, I yearn to make the world a better place while doing what I love.
和这些数学家一样，我渴望在做自己喜欢的事情的同时让世界变得更美好。
(Don’t we all?) Philosophically, I see the promise in applied category theory.
（我们不都是这样吗？）从哲学上来说，我看到了应用范畴论的希望。
Time will tell whether the approach will genuinely help humanity or the planet.
时间会证明这种方法是否能真正帮助人类或地球。
But for those who feel called to do good and to do math, it’s worth a try.
但对于那些觉得自己有义务做好事和做数学的人来说，值得一试。
Correction: March 9, 2026 An earlier version of the text described a one-dimensional vector space as a “category.” Rather, they are considered objects in a category in which the morphisms are linear maps.
更正：2026 年 3 月 9 日 该文本的早期版本将一维向量空间描述为“类别”。相反，它们被视为某个类别中的对象，其中态射是线性映射。
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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。