Have you ever wondered whether pure, crystalline mathematics — the kind scribbled on blackboards in university halls — could roll up its sleeves and help fix our broken planet? It sounds wildly unlikely. Math deals in abstractions, right? Numbers, proofs, theorems that float above the messy, tangled reality of living ecosystems and warming oceans. And yet, a growing community of mathematicians believes the most abstract branch of math ever created might hold the key to understanding our world in a profoundly new way.
Welcome to FreeAstroScience.com, where we break down complex scientific ideas into language that makes sense — because the sleep of reason breeds monsters, and we believe your mind should never stop asking questions. Today, we're taking you inside a story that reads like science fiction but is very real: the rise of applied category theory, a movement sometimes called "green math." Whether you're a science enthusiast, a curious student, or simply someone who cares about the future of our planet, stay with us until the end. This one's worth your time.
📑 Table of Contents
- 1. What Is Category Theory — And Why Should You Care?
- 2. John Baez and the Birth of "Green Math"
- 3. Why Nature Isn't a Machine — And Why That Matters
- 4. From Pandemics to AI Safety: Where Green Math Already Works
- 5. How Do Categories Actually Work? (A Simple Explainer)
- 6. What Challenges Stand in the Way?
- 7. What Does the Future of Green Math Look Like?
Can the Most Abstract Math Really Make Our World Better?
1. What Is Category Theory — And Why Should You Care?
Let's start at the beginning. Category theory was born in 1945 as a way to formalize relationships between mathematical objects. Numbers, functions, sets — these are all mathematical objects. But here's what makes category theory special: it doesn't care what an object is made of. It cares about how objects relate to each other.
Think of chess. A black king could be a carved wooden piece or a saltshaker — it doesn't matter. What defines the black king is how it moves on the board and how it interacts with other pieces . That's the category theorist's worldview. Relationships define identity.
A category, then, is a collection of objects plus the relationships between them. Mathematicians call those relationships morphisms — fancy word for "the ways things connect" . Picture a diagram with boxes and arrows. Boxes are objects. Arrows are morphisms. Category theorists study how to map, overlap, or connect different categories together.
Now, you might think this sounds painfully abstract. And you'd be right — it is abstract. It sits at the very top of the mathematical hierarchy, a bird's-eye view of how structures relate to one another. But that extreme generality? That's exactly what makes it powerful.
A Quick Everyday Example: Dollars vs. Feet
We all intuitively understand categories, even if we don't know the word. Consider this: you can multiply 5 feet by 3 feet to get 15 square feet. But $5 times $3? That's nonsense. There's no such thing as "square dollars" .
Dollar values live in what mathematicians call a one-dimensional vector space. You can add $5 and $3. You can multiply $5 by the pure number 3. But multiplying dollars by dollars isn't a valid operation in that category.
We manage to avoid these mistakes at the grocery store without any mathematical training. But when concepts get more varied and complex — say, in epidemiological modeling — things fall apart fast.
2. John Baez and the Birth of "Green Math"
In 2011, an influential mathematical physicist named John Baez wrote something unexpected on his blog. "I've spent a long time exploring the crystalline beauty of traditional mathematics," he said, "but now I'm feeling an urge to study something slightly more earthy" .
Baez splits his time between the University of California, Riverside and the University of Edinburgh. He'd grown increasingly worried about the state of the planet. And he had a bold idea: mathematicians could — and should — do something about it .
He called for the development of "green math" — new mathematics designed to capture the workings of Earth's biosphere and climate. His weapon of choice? Category theory, the very branch of abstract math where he'd built his career .
A cousin of the folk singer and activist Joan Baez, John was heavily influenced by his uncle (Joan's father), a physicist and socially active Quaker. Helping the world wasn't optional for him. It was, as he put it, "infused in me" .
Since that 2011 blog post, more than 100 mathematicians have joined Baez as "applied category theorists," attempting to model real-world systems in new ways. Applied category theory now has its own annual conference, an academic journal, and an institute — the Topos Institute in Berkeley, California. There's even a research program funded by the U.K. government .
That said, skepticism hasn't disappeared. "When I say we're underdogs and nobody likes us, it's not completely true, but it's a bit true," admitted applied category theorist Matteo Capucci
3. Why Nature Isn't a Machine — And Why That Matters
Here's where things get philosophical — and deeply important.
Baez argues that we've been misunderstanding biological systems for centuries. We treat them like machines: objects that take in matter and energy, produce outputs we want, and generate waste we ignore. "We focus on the part we care about and ignore the waste and where is the energy coming from," he said. "Our whole technology and indeed our whole mathematics is based on that attitude" .
But living systems aren't machines. Evolution has made life "incredibly subtle and complicated in ways we can't fully fathom" . Genes aren't neat, discrete parts with single purposes — each gene has numerous roles and impacts. In a healthy ecosystem, there's no waste at all. One creature's poop is another creature's feast don't think we have the math to understand such systems yet," Baez said .
That's a stunning admission from a mathematician. And it's a call to action. Baez believes that modeling these systems will require new categories with previously unstudied logical structures — mathematical frameworks that don't yet exist.
Why does this matter for the rest of us? Because the way we think about nature shapes the way we treat it. If we see forests and oceans as raw materials for our machines, we'll keep exploiting them until there's nothing left. "That attitude that we have right now is running into a wall," Baez warned. "That attitude winds up destroying the whole planet" .
Here's the hopeful twist. If we can develop math that sees nonhumans, ecosystems, and the climate as objects in a shared category — connected to us, interdependent with us — we might value them more . Not because we're told to, but because the math itself makes the relationships visible.
4. From Pandemics to AI Safety: Where Green Math Already Works
Applied category theory hasn't saved the biosphere yet. The applications aren't as "green" as Baez originally hoped. But the approach is showing real potential in some areas that affect all of us .
Fighting Pandemics with StockFlow
During Canada's pandemic response, computer scientist Nate Osgood at the University of Saskatchewan ran into a problem. The existing modeling software didn't allow experts from different fields to combine their knowledge effectively.
Epidemiologists often use stock-and-flow diagrams — illustrations with "stocks" of people (susceptible, infected, recovered, dead) and arrows showing flows between them based on factors like exposure or virulence . Stocks and flows are just objects and morphisms of a category. The arrangement of boxes and arrows translates directly into equations that describe how an outbreak evolves.
Over the last few years, Osgood, Baez, and their team have developed a software package called StockFlow that formalizes this kind of modeling. Different specialists can model different aspects of an outbreak — say, how health disparities affect infection rates — and these categories can be composed into larger ones .
"Category theory is able to handle those fancier forms of composition," Baez explained
Imagine a pandemic hitting tomorrow. Experts in virology, public health, economics, and social behavior each have their own data and models. Without a shared mathematical language, combining those models is like assembling IKEA furniture from five different instruction manuals — in five different languages. StockFlow gives everyone a lingua franca: category theory.
StockFlow hasn't spread widely among epidemiologists yet, but Osgood teaches it to his students. As mathematician Tom Leinster of the University of Edinburgh put it: "This is genuinely something that could be used. It's serious stuff" .
Safeguarding Artificial Intelligence
Here's where things get especially exciting — and urgent. Amar Hadzihasanovic of Tallinn University of Technology and Matteo Capucci are both part of Safeguarded AI, a project funded by ARIA (a U.K. government–funded advanced research agency). Their goal: apply category theory to AI safety.
The question they're tackling is this: how can unpredictable, error-prone AI systems be trusted to run essential real-world infrastructure — nuclear plants, power grids, hospitals?
Their 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"
5. How Do Categories Actually Work? (A Simple Explainer)
Let's go a bit deeper — but gently. Here's a practical example of how applied category theory works in the real world.
In a 2022 lecture, mathematician David Spivak (co-founder of the Topos Institute) described a scenario: an accountant tells a category theorist about the objects in their database — employees, dollar amounts, departments. The category theorist then builds a formal model — a category with a rigorous logical structure. That category can then be connected to other categories representing other databases and spreadsheets, creating a model of the entire company .
In other words, applied category theory is a universal language for talking about the different parts of some giant system . Different databases "speak" different languages. Category theory translates between them.
"David has a real imperative to formalize and make legible the world," said Brendan Fong, the co-founder and CEO of the Topos Institute. "The thing he hates most in the world is miscommunication" .
And if you're curious about how conventional modeling can go wrong without categorical thinking, consider Baez's epidemiology example. When you type "35" into a standard modeling program, the software doesn't know if it means 35 dollars, 35 people, or 35 doses of a drug. You're "conflating those all as just numbers, and that makes it easier to make mistakes" Category theory prevents those errors by keeping objects in their proper categories.
6. What Challenges Stand in the Way?
We'd be dishonest if we painted this as a smooth ride. Applied category theory faces real obstacles.
Climate modeling — one of Baez's original targets — remains out of reach. Climate models are already sophisticated enough to function, even though they lack mathematical rigor in how their parts fit together. Starting over with category theory would require enormous effort, and working scientists are understandably hesitant.
"It's one of the challenges we always face in applied category theory," said Hadzihasanovic. "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"'s also a political dimension. No amount of beautiful math can fix an inadequate political response to the climate crisis. Math can illuminate the structure of problems. It can't force people to act.
And then there's the perception issue. Category theory has a reputation for being impenetrable. Even within mathematics, it's sometimes dismissed as "abstract nonsense" (a half-affectionate, half-dismissive nickname). Convincing practical scientists to adopt it requires patience, clear communication, and working prototypes.
7. What Does the Future of Green Math Look Like?
Despite the challenges, there's a quiet confidence among applied category theorists. Systems are growing more complex and interconnected every year. AI is becoming more involved in managing critical infrastructure. "Winging it" with patchwork models that don't share a common logical foundation? That approach has limits.
"This is going to be, eventually, very important work," Hadzihasanovic said.
Many practitioners got into this field because they share Baez's environmental values and hope to tackle greener problems in time. Baez himself still carries that hope — the hope that new mathematics can help us "be kinder to the world if we understand the world a bit better" there's something almost poetic about his vision. If we can see ourselves and the natural world as objects in a shared category — bound by relationships, dependent on each other — then math isn't just an abstraction. It becomes an argument for empathy.
As columnist Natalie Wolchover wrote for Quanta Magazine: "Like these mathematicians, I yearn to make the world a better place while doing what I love. (Don't we all?)"
A Final Thought
So here's where we land. Applied category theory is still young. It hasn't solved climate change or ended pandemics. The skeptics have fair points. But a movement of over 100 mathematicians, an annual conference, a dedicated journal, government funding, and real software like StockFlow — these aren't pipe dreams. They're foundations.
What strikes us most at FreeAstroScience is the underlying philosophy. John Baez isn't asking math to do something math has never done. He's asking us to see differently. To stop treating nature as raw material. To recognize that the relationships between things — between species, between systems, between people — are what define them. Not what they're made of, but how they connect.
That's category theory. That's also, in a way, wisdom.
We hope this article gave you something to think about — something that stirs both your mind and your heart. Here at FreeAstroScience.com, we exist because we believe complex scientific ideas deserve simple, clear explanations. We want to educate you — not to fill your head with facts, but to make sure you never turn off your mind. Keep it active. Keep it curious. Because the sleep of reason breeds monsters.
Come back to FreeAstroScience.com soon. There's always more to explore.
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