内容来源：https://www.quantamagazine.org/can-the-most-abstract-math-make-the-world-a-better-place-20260304/

内容总结：

最抽象的数学能让世界更美好吗？

当数学物理学家约翰·贝兹在2011年发出呼吁，希望用高度抽象的“范畴论”来建模地球生物圈、发展“绿色数学”时，这听起来像是一个遥不可及的梦想。传统数学擅长描述简单孤立系统，而面对生态系统这般复杂巨系统，简洁的数学模型往往力不从心。

然而十余年来，已有超过百名数学家投身“应用范畴论”研究，试图以全新方式刻画现实世界。他们建立了年度会议、学术期刊、专门研究所以及获得英国政府资助的研究项目。尽管学界仍存广泛质疑，但这一领域已悄然在流行病学和人工智能安全等关键领域展现出实用潜力。

范畴论诞生于1945年，核心思想是通过“对象”及其间的“态射”（关系）来形式化数学结构。例如在流行病学模型中，“易感人群”“感染者”等群体可作为对象，而病毒传播过程就是连接它们的态射。应用范畴论提供了一套严谨框架，能避免传统建模中混淆“35人”与“35美元”这类单位错误，使复杂系统的描述更具逻辑一致性。

目前，研究团队已开发出用于流行病建模的软件工具StockFlow，帮助专家整合多领域知识；英国政府资助的“安全人工智能”项目则尝试用范畴论构建AI的可靠训练环境，以应对核电站、电网等关键系统的安全管理需求。

尽管在气候科学等更“绿色”的领域尚未取得突破——部分因为现有气候模型虽缺乏数学严谨性但已能运行，推倒重来成本过高——但研究者们坚信，随着系统复杂性与互联性日益增强，这种注重模块化组合的数学方法将愈发重要。

贝兹指出，当前人类常错误地将生命系统归类为“机器”，忽视其循环共生特性。他期待发展能真正理解生态系统的新数学，帮助人类超越“将自然视为原材料”的思维定式。“如果我们能更好地理解世界，或许就能更友善地对待它。”

应用范畴论能否真正造福人类与地球，仍需时间检验。但对于那些既热爱数学又渴望改善世界的研究者而言，这条探索之路本身已充满意义。

中文翻译：

最抽象的数学能让世界更美好吗？

引言

“我花了很长时间探索传统数学的晶体之美，但现在我渴望研究些更接地气的东西。”约翰·贝兹在2011年的博客中写道。这位在加州大学河滨分校与爱丁堡大学之间奔波的数学物理学家，对地球现状日益忧虑，并认为数学家们可以为此做些什么。

贝兹呼吁发展他称之为“绿色数学”的新数学，以更精准地描述地球生物圈与气候的运行机制。作为个人实践，他尝试运用自己精通的范畴论——这门高度抽象的数学分支——来为自然界建模。

这听起来像是天方夜谭。数学擅长描述简单孤立的系统，但从原子到生物体再到生态系统，随着系统日趋复杂，简洁的数学模型往往力不从心。

然而自贝兹发文以来，已有百余名数学家以“应用范畴论学者”的身份加入探索，尝试以全新方式为各类现实系统建模。如今应用范畴论已拥有年度会议、学术期刊、专门研究所以及英国政府资助的研究项目。

但质疑声依然不绝于耳。“说我们是弱势群体且不受欢迎，虽不完全准确，却也有几分真实。”应用范畴论学者马泰奥·卡普奇坦言。

我试图了解这个新兴领域：看似最纯粹的数学如何能破解生物圈这般复杂的系统？它是否显著优于其他建模方法？数学真能“变绿”吗？起初我并不乐观。

出乎意料的是，应用范畴论近期已取得若干成果。尽管应用方向尚未如贝兹所愿聚焦环保，但该方法已在流行病学与人工智能安全等重要领域展现潜力。最抽象的理想化模型，或许真能帮助我们理解最混沌的现实。

范畴论诞生于1945年，旨在形式化数学对象间的关系，随后迅速发展为数学中富有生命力的分支。

何为数学对象？数字、函数、集合皆属此列。在范畴论学者眼中，对象的本质在于其与他者的关联。国际象棋里的黑王是什么？“你可以说它是雕成特定形状并涂黑的木块，但这无关紧要——它甚至可以是盐瓶。”爱丁堡大学数学家汤姆·莱因斯特解释道。黑王的本质在于其在棋盘上的移动规则，以及如何擒获对方棋子或被将军。

范畴即对象及其关系（态射）的集合。以国际象棋为例构建范畴时，可用方框表示每个对象（合法棋局），再用箭头连接方框以表征态射（可行棋步）。范畴论学者研究如何映射、叠加或联结不同范畴。

我们其实都具备范畴的直觉认知。例如明白5英尺与5美元在数学上不等价：5英尺乘以3英尺可得15平方英尺，但5美元乘以3美元毫无意义（不存在“平方美元”）。你可以将5美元与3美元相加，或用5美元乘以数字3（而非货币单位）。美元与英尺分属不同范畴。

对范畴论学者而言，美元金额属于一维向量空间范畴的对象。想象一条数轴：金额如同从原点出发指向某处的箭头（向量）。两个向量可首尾相接实现加法，但向量乘法在一维向量空间中并非合法态射。

即便对向量空间或态射一无所知，我们也能在收银台避免尴尬的范畴错误。然而当概念比距离和货币更多元复杂时，问题便接踵而至。

“这在建模中屡见不鲜，比如流行病学建模。”贝兹指出，“用传统软件建模时，输入‘35’并不会区分这是35美元、35人还是35剂药物。这些不同范畴被混同为数字，更易导致错误。”

应用范畴论提供了以对象与态射为框架的现实系统建模方法。“范畴是组织逻辑结构的方式。”致力于范畴论应用的伯克利拓扑研究所联合创始人兼CEO布伦丹·方如是说。

物理学家鲍勃·科克在21世纪初将其应用于量子力学，后扩展至量子计算推理。数年后，贝兹开始思考生物圈的范畴化，而拓扑研究所联合创始人、数学家戴维·斯皮瓦克则通过数据库研究独立开创了应用范畴论。“戴维始终致力于将世界形式化、清晰化。”方评价道，“他最痛恨的就是沟通错位。”

在2022年的线上讲座中，斯皮瓦克描绘了应用范畴论的实践场景：会计师向范畴论学者说明数据库中的对象（如员工、金额），学者据此构建形式化模型（具备严谨逻辑结构的范畴），再通过连接其他范畴（对应不同数据库与表格）来模拟整个公司运作。应用范畴论由此成为描述巨系统异构部分的通用语言。

气候建模——贝兹最初设想的绿色数学应用——试图模拟地球这个典型巨系统。不同领域的专家需要以逻辑方式整合知识与数据流。但贝兹等人坦言，应用范畴论在气候科学中尚未立足，部分因为现有气候模型虽缺乏数学严谨性，却已足够复杂实用。数学家们认为，严谨性能增强模型的鲁棒性、灵活性及信息整合能力，但推倒重来既需说服他人，也要付出巨大努力。

“这是应用范畴论始终面临的挑战。”爱沙尼亚塔林理工大学的阿马尔·哈兹哈萨诺维奇解释道，“我们可以建议他人‘按基本原理重构模型会更好’，对方则会问‘这要花多久？’在收获成效前，这是巨大的投入。”

数学虽难改观气候危机的政治应对困局，但应用范畴论已在其他公共关切领域取得进展。

例如，贝兹正与拓扑研究所及加拿大计算机科学家内特·奥斯古德合作。奥斯古德在萨斯喀彻温大学参与加拿大疫情防控时，因现有建模软件无法整合多领域知识而深感挫败。

流行病学家常用存量-流量图预测疫情发展：以方框表示人群存量（易感者、感染者、康复者、死亡者），箭头表示基于暴露或毒力等因素的流量变化。存量与流量正是范畴中的对象与态射，图中方框与箭头的排布可转化为描述系统演化的方程。

过去几年间，奥斯古德、贝兹及其团队开发了名为StockFlow的软件包，将此类建模形式化。专家可分别构建疫情不同侧面的范畴（如健康差异如何影响易感人群感染率），再组合成更大范畴。“范畴论擅长处理这类复杂组合。”贝兹说。

StockFlow尚未在流行病学家中普及，但奥斯古德已将其纳入教学，以期培养新一代建模者。“这确实是具有实用价值的工作。”莱因斯特评价道，“是严肃的成果。”

与此同时，哈兹哈萨诺维奇与卡普奇共同参与了由英国政府资助的先进研究机构ARIA支持的“人工智能安全防护”项目，尝试用范畴论解决AI安全问题：如何让不可预测且易出错的AI系统可靠地管理核电站或电网等关键现实系统？

这个团队的解决方案颇具巧思：为AI构建与现实系统逻辑同构的复杂系统形式化模型，精确表征各类对象间的态射关系。

“范畴论提供了模块化、组合化的实现路径。”卡普奇表示，“我们正在开发可广泛应用于多种场景的基础技术。”

随着系统日益复杂互联、AI更深介入人类事务，应用范畴论学者相信他们的方法终将彰显价值。“临场发挥终非长久之计，”哈兹哈萨诺维奇断言，“这终将成为极其重要的工作。”

许多研究者投身此领域，既因认同贝兹的环保理念，也期待未来解决更绿色的课题。贝兹仍怀热望——这位民谣歌手兼活动家琼·贝兹的表亲，深受同样身为物理学家与社会活动家的叔叔（琼·贝兹之父）影响。“造福世界而不仅追求自我享乐，这种信念已融入我的血脉。”

当我问及范畴论如何帮助理解生物圈时，贝兹指出人类对生物系统的范畴化存在根本谬误：我们误将其视为机械——这种通过输入物质能量、输出目标产物与废料来完成特定任务的对象。“我们只关注所需部分，却忽视废料去向与能量来源。”贝兹说，“我们的整个技术体系乃至数学基础都建立在这种认知上。”

生命系统属于截然不同的范畴。它们并非为执行任务而构建，进化使生命“呈现出超乎想象的精妙与复杂”。例如基因并非机械中功能单一的零件，每个基因都具有多重角色与影响。在生态系统中没有废料，一种生物的排泄恰是另一种生物的盛宴。

“我认为我们尚未掌握理解此类系统的数学工具。”贝兹相信建模这类系统需要发展具有全新逻辑结构的新范畴，“这正是我想探索的数学。如果我们能更好地理解世界，不再将自然视为机器掠夺的原材料，或许能更温柔地对待地球。当前这种认知方式已撞上南墙，最终将摧毁整个星球。”

的确，当我们将非人类生命、生态系统乃至气候与自身置于同一范畴中构想时，或许能更珍视彼此的存在。

与这些数学家一样，我渴望在热爱的事业中让世界更美好。（谁不是呢？）从哲学层面看，应用范畴论蕴藏着希望。它能否真正造福人类与地球，时间将给出答案。但对于那些既怀济世之心又爱数学之美的人而言，这值得尝试。

英文来源：

Can the Most Abstract Math Make the World a Better Place?
Introduction
“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. 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.
Baez called for the development of new mathematics — he called it “green” math — to better capture the workings of Earth’s biosphere and climate. 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.
It sounds like a pipe dream. 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. The systems are just too complex.
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. 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.
Skepticism abounds, however. “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.
I set out to learn what this burgeoning research area is about. How could one of the seemingly most rarefied realms of pure math help demystify a system as complex as the biosphere? Is it a significant improvement on other approaches to modeling? Can mathematics really be green? It didn’t seem promising.
To my surprise, I’ve learned that applied category theory has had some wins lately. 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. It seems plausible that the most abstract idealizations can help make greater sense of the messiest realities.
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.
What do we mean by mathematical objects? Numbers, functions, and sets are examples. To a category theorist, what defines an object is its relationships to others. What is a black king in chess? “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. 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.
A category is a collection of objects and these relationships, or morphisms. Let’s consider that chess set as a category. 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. Category theorists study how to map, overlap, or connect various categories.
We’re all intuitively aware of categories. We know, for example, that 5 feet and $5 are not the same mathematically. You can multiply 5 feet by 3 feet to get 15 square feet. But you can’t multiply $5 and $3 — there’s no such thing as square dollars. You can add $5 and $3, or you can multiply $5 by 3 (the number, not dollars). But $5 times $3 is meaningless.
To a category theorist, dollar values are the objects in a category called a one-dimensional vector space. Picture the number line; a dollar amount is like an arrow (or “vector”) anchored at zero that reaches some distance along the line. You can add two vectors by placing them tip to tail, but multiplying vectors isn’t a valid morphism in a one-dimensional vector space.
Despite knowing nothing of vector spaces or morphisms, we somehow manage to avoid making embarrassing category errors at the checkout counter. But when concepts are more varied and complex than distances and dollars, we run into problems.
“It shows up all the time in modeling, for example epidemiological modeling,” Baez told me. “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. And so you’re conflating those all as just numbers, and that makes it easier to make mistakes.”
Applied category theory provides a framework for modeling real-world systems in terms of objects and morphisms. “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.
The physicist Bob Coecke applied it to quantum mechanics in the 2000s, which has since been extended to reasoning about quantum computation. 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. “David has a real imperative to formalize and make legible the world,” Fong said. “The thing he hates most in the world is miscommunication.”
In a 2022 lecture I watched online, Spivak envisioned how applied category theory might work in practice. An accountant tells an applied category theorist about the objects in their database, such as employees and dollar amounts. 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. In this way, applied category theory is a lingua franca for talking about the heterogeneous parts of some giant system.
Climate modeling — one of Baez’s initial candidates for green mathematics — attempts to simulate the archetypal giant system: Earth itself. Experts in different parts of the Earth system must assemble their knowledge and streams of data in a logical way to understand the whole. 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. 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.
“It’s one of the challenges we always face in applied category theory,” said Amar Hadzihasanovic of Tallinn University of Technology in Estonia. “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.”
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.
For example, Baez has been collaborating with Topos and a Canadian computer scientist, Nate Osgood, who specializes in epidemiological modeling of disease outbreaks. 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.
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. Stocks and flows are just objects and morphisms of a category. The arrangement of boxes and arrows in the diagrams translates into equations describing the system’s evolution.
Over the last few years, Osgood, Baez, and their team have developed a software package called StockFlow that formalizes this kind of modeling. 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. “Category theory is able to handle those fancier forms of composition,” Baez said.
StockFlow has yet to spread among epidemiologists, but Osgood teaches it to his students in hopes of inoculating the next generation of modelers. “This is genuinely something that could be used,” said Leinster. “It’s serious stuff.”
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. 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?
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. These models must have the same logical structure as the real system, correctly representing the morphisms between many different types of objects.
“Category theory gives you a modular and compositional way of doing this,” Capucci said. “We are developing fundamental technology that we can deploy in so many situations.”
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. Winging it won’t cut it. “This is going to be, eventually, very important work,” Hadzihasanovic said.
Many practitioners got into the field because they share Baez’s environmental ethos and hope to take on greener problems in time. Baez still has high hopes. 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. It’s “infused in me,” he said, to help the world and “not just enjoy myself.”
I asked what it is about the biosphere that he thinks category theory can help us understand.
In his view, we improperly categorize biological systems. We mistake them for machines, objects that perform specific tasks by taking in matter and energy and producing desired outputs and waste. “We focus on the part we care about and ignore the waste and where is the energy coming from,” Baez said. “Our whole technology and indeed our whole mathematics is based on that attitude.”
Living systems are a different category, though. They’re not built to perform tasks. Evolution has made life “incredibly subtle and complicated in ways we can’t fully fathom,” he said. Genes aren’t discrete parts of a machine with their own purposes, for instance; they all have numerous roles and impacts. In an ecosystem, there’s no waste; one creature’s poop is another’s feast.
“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. “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. That attitude that we have right now is running into a wall. That attitude winds up destroying the whole planet.”
Indeed, we might value nonhumans, ecosystems, and the climate more by conceiving of them and ourselves as objects in a shared category.
Like these mathematicians, I yearn to make the world a better place while doing what I love. (Don’t we all?) Philosophically, I see the promise in applied category theory. Time will tell whether the approach will genuinely help humanity or the planet. But for those who feel called to do good and to do math, it’s worth a try.