Beauty is a dangerous word in science.
It's dangerous because it's slippery, because it resists measurement, and because the moment you try to pin it down, it shapeshifts into something else entirely. I know this feeling well — not just from years of studying physics and astronomy, but from a life spent navigating a world that wasn't designed for my body. When you live with dystonia, when your wheelchair is your constant companion through lecture halls in Bologna and seminar rooms in Milan, you develop a peculiar sensitivity to elegance. You learn to spot it in unexpected places. A ramp with the right incline. A proof with the right structure. A sentence that lands exactly where it should.
So when I read a new essay exploring whether mathematical beauty is real — or just a human "wow" becoming redundant in an age of AI — I felt something stir. Not just intellectual curiosity. Something deeper, almost physical, like the hum of a tuning fork pressed against bone.
The Scene in Bloomsbury
The essay, published on Aeon, opens with a vivid scene: a hot July day in London, the author heading to the Hardy Lecture at the London Mathematical Society. Emily Riehl of Johns Hopkins University is speaking about infinity category theory — a mathematical language so abstract it makes quantum mechanics look like a children's book. The question on the table: can we teach this language to computers so they can verify proofs and construct complex structures?
A few seats away sits Kevin Buzzard of Imperial College London, a man known for his colourful trousers and his blunt rejection of beauty in favour of rigour. Buzzard works on a computer proof assistant called Lean, and his mission is straightforward — convince every mathematician alive to write their proofs in it. "I reject beauty, I want rigour," he once said during a Q&A.
That sentence haunts me. Not because it's wrong. Because it's incomplete.
What Beauty Is Not
Let me simplify something here for those who don't spend their weekends reading topology papers (and honestly, good for you). The essay uses the map-colouring problem as a case study — a classic question in graph theory that asks: what's the minimum number of colours you need to paint a map so that no two neighbouring countries share the same colour?
You can solve this by brute force. Try every single combination. Check them all. It works. It's correct.
No mathematician will call it beautiful.
The author puts it perfectly: trying out all combinations is "plodding and pedestrian" . It doesn't offer a new perspective. It doesn't spark insight. It's the mathematical equivalent of writing "It rains" when you could write Longfellow's The Rainy Day. Both describe the same phenomenon. Only one makes you feel something.
This distinction matters more than most people realise. Mathematicians don't just prove theorems once — they prove the same theorem over and over again, searching for a more elegant path. It's not repetition. It's refinement. It's the difference between photographing a mountain and painting it from a new angle each time, watching how the light changes.
Erdős, God, and THE BOOK
The great Hungarian mathematician Paul Erdős believed that God — in whom Erdős himself didn't believe — keeps a volume containing the perfect proof for every theorem . He called it THE BOOK. You don't need to believe in God, Erdős argued, but as a mathematician, you should believe in THE BOOK.
I love this idea. I love it because it's playful and serious at the same time. It connects to Plato's Theory of Forms — the notion that somewhere out there exists a perfect version of every imperfect thing we encounter. We never reach those perfect forms, but we inch closer. Every proof, every theorem, every late-night scribble on a napkin is an attempt to close the gap between what we've written and what THE BOOK already contains.
Erdős died before completing his own version of this volume. It was published posthumously in 1998 as Proofs from THE BOOK. Mathematicians still boast about their "Erdős number" — their degrees of separation from him through collaboration . It's academic genealogy meets celebrity culture, and it's wonderful.
The Three Pillars: Simplicity, Surprise, Vitality
Here's where the essay gets genuinely interesting — and where I want to slow down, because the ideas deserve it.
The author proposes a definition of mathematical beauty built on three pillars. I'm going to simplify these for a general audience, because that's what we do at FreeAstroScience — we take the complex and make it accessible without stripping away the wonder.
The first pillar is simplicity. Not simplicity of content, but simplicity of expression. Think of it like poetry: the test lies in capturing a vast image through the precise selection of a concise sequence of words. A proof can describe something enormously complicated, but the structure of the argument itself should feel clean, transparent, and even . The author draws a sharp line here: while ambiguity is praised in poetry, in mathematics it's "disgraced". A deliberately ambiguous mathematical structure isn't beautiful. It's a poem pretending to be a proof.
The second pillar is surprise. This one's harder to formalise, and the author admits it . Surprise in mathematics happens when someone borrows a technique from an entirely different field — say, using geometry to solve a problem in algebra, or importing ideas from quantum gravity to prove theorems in topology . It's the intellectual equivalent of a plot twist you didn't see coming but, in hindsight, was inevitable.
The third pillar is vitality. This is my favourite, and the one that resonates most with my own experience. The author borrows the concept from Cecil Day-Lewis's The Poetic Image (1947), arguing that good mathematicians — like good poets — have high vitality. They live in the present moment. They observe with a fresh eye. They don't, as the literature scholar John Livingston Lowes put it, "ensconce themselves like hermit-crabs, generation after generation, in the cast-off shells of their predecessors" .
And here's the kicker: sometimes it's a fresh PhD student, just out of college, who spots the proof from THE BOOK — not the seasoned professor who's been teaching the same material for decades. That's vitality. That's the living, breathing quality that makes a mathematical structure move, excite, and create.
Russell's Paradox and Fermat's Stubbornness
The essay offers two brilliant examples that sit at opposite ends of the simplicity spectrum.
Russell's paradox is simple in both statement and proof . Imagine a set R that contains all sets that don't contain themselves. Does R contain itself? If yes, then by definition it shouldn't. If no, then by definition it should. Contradiction. Such a set can't exist — and therefore, a universal set containing everything, including itself, doesn't exist . The proof takes a few lines. The implications shook the foundations of mathematics and spawned entire new fields like category theory and type theory .
Fermat's Last Theorem is the opposite: the statement is so simple a 10-year-old can understand it (and indeed, a 10-year-old Andrew Wiles did) . No three positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for any integer n greater than 2. That's it. But proving it required Wiles — and generations of mathematicians before him — to invent entirely new techniques in algebraic number theory. Even today, the beauty of the proof is appreciated by only a few .
Both are beautiful. Both are simple. Both surprise. Both carry vitality. The difference is where the complexity lives — in the idea itself or in the machinery required to reach it.
The Four-Colour Theorem and the Machine
Now we arrive at the uncomfortable part.
The four-colour theorem — the conjecture that any map can be coloured with no more than four colours such that no neighbouring countries share a colour — remained open for over a century . It was finally proved in 1977, and the proof required a computer . A program found all possible configurations, converted them into networks, assigned numerical charges to each node, and shuffled those charges around according to specific rules. If a map truly needed more than four colours, the numbers wouldn't balance against Euler's characteristic formula .
It's correct. It's verified. It's accepted.
Is it beautiful? The author asked many senior mathematicians this question. The answers were "not straightforward and were surprisingly distinct".
I find this fascinating. The four-colour theorem proof was the beginning of computer-assisted proofs, and now, nearly fifty years later, we're asking whether AI can go further — not just verify, but create mathematics. The author notes that interest in AI-driven mathematics has been "exponential," with many mathematicians leaving traditional research to explore its potential. A group of distinguished mathematicians recently designed 10 research-level questions for AI to tackle, and at the time of writing, various AI companies had claimed solutions under evaluation.
Can a Machine Have Vitality?
This is the question that keeps me up at night — or at least keeps me scrolling past midnight, which for someone with my condition amounts to the same thing.
The author's definition of mathematical beauty is elegant: "A simple mathematical structure that surprises even the most experienced mathematicians and transfers a sense of vitality" . And then comes the honest admission: "But is an AI-assisted proof simple or surprising? How do we define vitality in a machine? On these questions, the jury is out" .
I'm torn too. And I say this as someone who's spent years building FreeAstroScience into a community of tens of thousands of followers, many of whom are young people discovering science for the first time through digital tools. I believe in technology. I believe in its power to democratise knowledge. My wheelchair is itself a piece of technology that gives me access to a world that would otherwise be closed off.
But vitality? That's something else entirely.
The author wonders whether our limbic system — the emotional core of the brain — is required for mathematical creativity . "Can we write proofs without emotional kicks?" they ask. "I am also unsure if perfectly efficient brains can come up with novel revolutionary ideas".
I think about my own journey through physics and astronomy. The moments of genuine insight — the ones that changed how I see the universe — were never purely rational. They were tangled up with frustration, with exhaustion, with the physical sensation of my hands shaking as I wrote equations on a whiteboard during my Erasmus semester at Sabancı University in Istanbul. The insight came through the struggle, not despite it.
Can a machine struggle? Can it feel the weight of a problem it can't solve? Can it experience the electric jolt of suddenly seeing a connection that wasn't there before?
I don't think so. Not yet. And "not yet" is doing a lot of heavy lifting in that sentence.
The Real Question Behind the Question
The essay ends with a quietly devastating observation. Some mathematicians say they prefer "truth and only truth." But when shown a beautiful piece of mathematics, most "immediately recognise, enjoy and even wholeheartedly smile" at it.
"In fact, they spend their whole lives in search of one".
That line hit me like a chord change in a song you thought you knew. Because it reveals something the rigour-only crowd doesn't want to admit: the search for beauty isn't a distraction from truth. It's a compass pointing toward it. The most beautiful proofs tend to be the most illuminating ones. Simplicity, surprise, and vitality aren't decorations — they're signals that you've found something deep.
This debate — truth versus beauty, rigour versus elegance, human versus machine — isn't just about mathematics. It's about what we value in the act of thinking itself. If we hand over the creative process to AI, if we let machines generate proofs that are correct but opaque, efficient but lifeless, we won't just lose beauty.
We'll lose the reason most mathematicians became mathematicians in the first place.
Never Give Up on Wonder
I've spent my life proving that limitations don't define you. That a body in a wheelchair can house a mind that reaches for the stars — literally, in my case. And I've learned that the most important quality in any pursuit, whether it's science or art or simply getting through a difficult day, is vitality. The willingness to stay present. To look at old problems with fresh eyes. To refuse the cast-off shells of your predecessors.
AI will get better at mathematics. It will verify more proofs, solve more problems, and generate structures we haven't imagined. That's exciting. That's progress.
But beauty? Beauty requires a heartbeat. It requires the messy, imperfect, emotionally charged experience of being human — of sitting in a room in Bloomsbury on a hot July day, staring at a plaque bearing Hardy's name, and wondering whether the future of mathematics still has room for wonder.
I believe it does. I believe it must.
And if it doesn't — well, that's a problem no algorithm can solve.
Gerd Dani is the President of Free AstroScience. He writes from his wheelchair, and he never gives up.
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