Mar 5, 2026 08:17 PM
https://www.quantamagazine.org/can-the-m...-20260304/
INTRO: 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... (MORE - details)
INTRO: 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... (MORE - details)
