Can the Most Abstract Math Make the World a Better Place?

March 4, 2026Celsius Pictor for Quanta MagazineColumnistMarch 4, 2026“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.In 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. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.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.

Get Quanta Magazine delivered to your inbox Jeremy Howard via flickr, modified by QuantaA 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.”Xenja Santarelli, modified by QuantaIn 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.Robert Koorenny, modified by QuantaTo 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.ColumnistMarch 4, 2026 Get Quanta Magazine delivered to your inbox Get highlights of the most important news delivered to your email inbox Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected. Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.