The eye of the mathematician

Rita ahmadi

Aeon

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The eye of the mathematician

Is mathematical beauty real? Or is it just a subjective, human ‘wow’ that is becoming redundant in an AI age?

Einstein equations on a blackboard at Mount Wilson Observatory in California. Courtesy Caltech

is a stipendiary lecturer in mathematics at Mansfield College, University of Oxford. She did her DPhil at the University of Oxford.

Edited byRichard Fisher

It is a hot July day in London and I take the bus to Bloomsbury. I often come here for the British Library, the British Museum or the London Review Bookshop. More than a location, Bloomsbury feels like stepping into a work of art – maybe one of Virginia Woolf’s stories, or Duncan Grant’s paintings.

This time, I am here for mathematics: the Hardy Lecture at the London Mathematical Society (LMS), named after G H Hardy, a professor of mathematics at the University of Cambridge, a member of the Bloomsbury Group, and a president of the LMS. You may know him from the film The Man Who Knew Infinity (2015), in which he’s played by Jeremy Irons.

The 2025 lecture is by Emily Riehl of Johns Hopkins University in Baltimore, who is talking about a complex mathematical ‘language’ known as infinity category theory: could we teach it to computers so that they could understand it? If successful, computer programs could verify proofs and construct complex structures in this area.

A few seats to my left, I recognise Kevin Buzzard, wearing the multi-coloured, patterned trousers he’s known for among mathematicians. Based at Imperial College London, Buzzard is working on a computer proof assistant called Lean. His interest is personal: after long disputes with a colleague over a flawed proof, he lost trust, as he often puts it, in ‘human mathematicians’. His mission now is to convince all mathematicians to write their proofs in Lean. In the Q&A after one of his talks, he said of the debate between truth and beauty in mathematics: ‘I reject beauty, I want rigour’ – though his vibrant sense of fashion suggests otherwise.

Interest in an AI-driven approach to mathematics has been exponential, and many mathematicians have left traditional academic research to explore its potential. Recently, one group of distinguished mathematicians designed 10 active, research-level questions for AI to tackle. At the time of writing, various AI companies and researchers had claimed to find solutions, which were under evaluation by the community.

Sitting in the room in Bloomsbury, I stared at the Hardy plaque and wondered: would Hardy find proofs generated by AI beautiful? I wasn’t sure. He believed there should be a strong aesthetic judgment in mathematics, drawing parallels with poetry, and argued that beauty is the first test of good mathematics. He went as far as to say that there is no permanent place in the world for ugly mathematics.

If asked, many mathematicians today still talk about the aesthetic appeal of one approach over another.

Yet we live in a different century to Hardy and his Bloomsbury peers, with different technologies and techniques, so perhaps we need a clearer definition of what mathematical beauty actually is. Over the history of mathematics, we can find examples where both rigour and the pursuit of beauty have shaped mathematics itself. So, if we’re completely replacing this with a computer-assisted quest for truth and rigour, we ought to know what we’d be abandoning, if anything. Is mathematical beauty like the beauty in literature and art – or is it something else?

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It’s often hard to define a notion, particularly beauty. It seems easier to address what it is not than what it is. You don’t become fully enlightened, but you at least see a light in the darkness. So, let’s begin the abstract task of describing ‘what it is not’ with an example of a not-so-beautiful proof of the so-called ‘graph-colouring problem’.

Colouring problems are part of graph theory, a branch of mathematics that studies questions concerning configurations of nodes and the edges between them. Many complex problems in everyday life or science can be intelligently reduced to a graph with certain properties.

The map-colouring problem is perhaps the most famous example. The question asks for the minimum number of colours required to colour a map of countries such that no two countries sharing a border have the same colour. To see this explicitly, take a graph with four nodes – is it possible to colour the faces of this graph with two colours? If not, how many colours are at least necessary?

Graphic supplied by the author

Since the graph has four nodes, you can try all possible combinations. It is impossible to colour them with only two colours, because two triangles inevitably need to be coloured the same, which is a violation due to the shared edge. If you increase the number of nodes or edges and ask the same question, you can do the same – try all combinations. You may need many sheets of paper, colours and days, and perhaps a few assistants to check whether you have considered all possible combinations. Regardless of efficiency and despite the lack of elegance, in principle, it is possible.

That is why mathematicians prove the same theorem over and over again: they are looking for a beautiful proof

All mathematicians will nod to this proof as correct; depending on the skill of the problem-solver, they may even say: ‘Well done!’ However, no one, dead or alive, will smile at this proof or say: ‘Wow!’ None of these reactions is intentional – it is a matter of taste. Trying out all combinations is plodding and pedestrian. It doesn’t offer a new perspective or establish an insightful idea or technique. Mathematicians don’t find this proof elegant or beautiful or sublime, unless an inductive approach on the number of nodes and edges reveals a pattern or sparks an insight. Even then, the induction itself doesn’t signify any beauty, but the insight, discovered pattern or future perspective built upon it might.

The same goes for English speakers who don’t find beauty in the simple factual sentence ‘It rains,’ but do in Henry Wadsworth Longfellow’s poem ‘The Rainy Day’ (1841):

My life is cold, and dark, and dreary;It rains, and the wind is never weary;

In mathematics, you are looking for an extraordinary image. You don’t want to see ordinaries. Even if you use an ordinary method or image to prove the existence of a claim, you are not ultimately satisfied. That is why mathematicians prove the same theorem over and over again: they are looking for a beautiful proof. This attempt is different from copying The Last Supper or sculpting David many times. The phenomenon – rain – is the same, but each poem paints a different image or reveals a different angle, and therefore evokes different sensations.

Hardy was not the only one to see the poetry. The 20th-century Hungarian mathematician Paul Erdős also believed in aesthetic judgment in mathematics, arguing that God maintains the perfect proofs for all theorems in a volume Erdős called ‘THE BOOK’. A non-believer himself, Erdős used God as a metaphor in a 1985 lecture, saying you don’t need to believe in God but, as a mathematician, you should certainly believe in THE BOOK. Erdős died before completing his own version of this hypothetical volume, but the majority of the proofs were selected or rewritten by him, and published posthumously as Proofs from THE BOOK (1998). Erdős’s book therefore was his characterisation of elegant and beautiful proofs, as if a higher hand had already written them and we’re tasked only with discovering them.

Mathematicians and physicists often boast about their ‘Erdős number’, defined by their degrees of separation from him in terms of collaboration. Perhaps it not only shows one’s proximity to Erdős himself but also to the one who, in legend, had access to an ultimate version of THE BOOK.

Erdős’s perspective is connected to Plato’s Theory of Forms. Contrary to the subjective theory of aesthetics, this view describes a perfect world that contains all the beauties and Platonic ideals of our imperfect world. We never reach these perfect forms, but can get closer through our theories, objects, proofs and everything we can imagine.

Therefore, the discussion of beauty should not be limited to proofs. There are, after all, other mathematical techniques, insights or theorems, which I collectively call mathematical structures.

For example, a claim is formally called a conjecture. And when one can provide evidence, or formally a proof, for a conjecture, then it becomes a theorem, and if it is not a significant theorem, it is called a lemma. Some theorems remain unsolved and, hence, they remain as conjectures. If the mathematics community assesses that proving a conjecture is difficult, it becomes an open problem.

What if, instead of a proof, one could find a beautiful conjecture, algorithm or strategy for the map-colouring problem? The problem is one of the most dramatic in the history of mathematics, with a few notable ups and downs. In 1852, Francis Guthrie, while colouring a map of England, conjectured that to colour any map such that no two countries sharing a border have the same colour, we need no more than four colours – ie, the four-colour conjecture.

The conjecture was passed to Augustus De Morgan – the LMS building is named after him – by Guthrie’s brother, who was studying at University College London. De Morgan mentioned it to William Rowan Hamilton, and Hamilton to Arthur Cayley. Cayley eventually announced the problem at the LMS. In 1879, Alfred Kempe published a paper that claimed to establish the result. This led to Kempe being elected to the Royal Society and later becoming president of the LMS.

In 1890, the proof turned out to be wrong. But not everything wrong is useless. Percy John Heawood identified the flaws in Kempe’s argument and established a proof of the five-colouring problem using Kempe’s techniques. The four-colouring problem remained open for a century, until it was finally proved in 1977 using computer software. That marked the beginning of computer-assisted proofs.

Simplicity does not evoke the same meaning for mathematicians, even those who work in the same field

The proof uses a ‘discharging method’: essentially, a computer program finds all possible configurations, converts them into a network of dots connected by lines, then assigns numbers (charges) to each dot, and moves these numbers around according to certain conditions. The key insight is that if a map really did need more than four colours, these numbers wouldn’t add up properly when checked against Euler’s characteristics formula, a fundamental equation useful to analyse networks. The algorithm can be made more sophisticated and efficient, but all variants need, to some extent, computer involvement.

Do you consider this a beautiful proof? Over the past few months, I have asked many senior mathematicians the very same question, and whether they have certain criteria or a rule-book for deciding and discovering beautiful mathematical structures. The answers were not straightforward and were surprisingly distinct. But most mentioned that a beautiful piece of mathematics should be simple. ‘What do you mean by simple?’ I asked many times, and that was one point of divergence. Simplicity does not evoke the same meaning for mathematicians, even those who work in the same field. Some mean the number of lines in a proof; some mean its self-sufficiency, such as requiring few citations to other lemmas; some mean the simplicity of the proof idea or even of the theorem statement. For example, is the central idea simple enough to be explained to – and understood by – a layperson, even if the techniques and full proof are complicated and require training and mathematical knowledge?

Simplicity, I argue, is the first taste of beauty, for the same reason that Hardy believed mathematics is similar to poetry, not prose. In this sense, simplicity does not oppose the depth of a mathematical idea. But, like poetry, the test lies in capturing and describing an image through the precise selection of a sequence of words in a concise yet profound structure. The simplicity of a mathematical proof is the expression of the sequence of techniques, not the image itself. The image can be complicated yet transparent, as I shall clarify.

While in poetry ambiguity is praised, in mathematics it is disgraced. A mathematician does not attempt to produce an ambiguous piece of mathematics; in fact, transparency represents one litmus test of the right level of simplicity in mathematics. One aim of simplicity is to reduce ambiguity and to produce transparency. By ambiguity, I don’t mean the inadequacy of the reader. The layperson’s claim of ambiguity in Andrew Wiles’s 1995 proof of Fermat’s Last Theorem is not a verdict, as the reader is inadequate in this case. But a simple structure purposely designed to be ambiguous is a poem, not a piece of mathematics.

What version of simplicity? Is it the statement, the idea, or the execution of techniques? Perhaps all play some part; a beautiful structure may, to some extent, represent all of them. But not everything simple is considered beautiful. Returning to the poetry analogy, not every simple sequence of simple words creates a beautiful poem. In The Poetic Image (1947), Cecil Day-Lewis said that a sequence of innocent or apparently irrelevant words might elicit the strongest feelings, depending on the image and metaphor it creates. He brought up Robert Browning’s poem ‘Up at a Villa – Down in the City’ (1855):

The wild tulip, at end of its tube, blows out its great red bellLike a thin clear bubble of blood, for the children to pick and sell.

Day-Lewis admired the skill and the transformation of glass-blowing into a sensuous picture; but for him the image suddenly shifted to another, more emotional one when he heard ‘the children’ – not only the image but also the nature of the reactions and emotions suddenly change. The word ‘children’ is simple and innocent, yet its juxtaposition within this image transforms it into something complex, an incongruent picture that evokes different stages of emotional response. In this case, the poetic image is profound, layered and emotionally tense, while the structure and the words remain simple.

In mathematics, a similar situation can occur: a series of simple lemmas or definitions can produce a complex and beautiful idea or image. Take Russell’s paradox – identified by Bertrand Russell in the early 1900s – which states that a ‘universal set’ does not exist. This paradox is significant and shook the foundations of mathematics. Volumes have been written on its philosophical implications, and alternative foundational meta-mathematical languages were developed partly in response to it – for example, category theory and type theory. In this case, both the statement and the proof are simple, and one need know only a few intuitive notions to grasp it. However, the idea reveals a much deeper and more fundamental image.

Thus, a set defined in this way – one that contains all sets that do not contain themselves – cannot exist

It goes as follows. In mathematics, a set is a collection of things that sometimes share a common property. Most sets we think about aren’t members of themselves. For example, the set of all cats is not itself a cat, so it doesn’t contain itself.

Let us imagine that there exists a set, called R, which contains all sets that do not contain themselves. In mathematical notation, it looks like this:

But what about R itself? Does R contain R, or not? The answer is neither!

We can prove, by contradiction, that such a set cannot exist. First, assume that R is a member of itself. By definition, every member of R must not contain itself, so this can’t be true. But it’s also not possible that R is not a member of itself. In that case, R satisfies the defining condition for being included in R – it is a set that does not contain itself. Therefore, R must be in R, which again contradicts the assumption.

Thus, a set defined in this way – one that contains all sets that do not contain themselves – cannot exist. Consequently, a universal set that contains all sets, including itself, does not exist.

The opposite happens when a mathematical statement or theorem is simple, but the proof or the ideas and techniques behind it are complex. An example is Fermat’s Last Theorem. It states that no three positive integers – a, b, c – satisfy the following equation for any integer n>2:

This is so simple that the 10-year-old Andrew Wiles could read it, understand it, and even dream about solving it for the rest of his life. But it took him – and many mathematicians before him – years of learning and inventing new, highly complex techniques in algebraic number theory to solve it. Even today, the beauty of Wiles’s proof is appreciated by only a few.

Both Russell’s paradox and Fermat’s Last Theorem are not only simple, they take us by surprise – an element of which is the second feature of beauty. Similar to simplicity, surprise doesn’t have a fixed interpretation. I believe it’s even harder to formalise. Mathematicians working in a particular field become accustomed to certain proof techniques, lemmas and basic theorems. This is essentially their main toolbox. To put it rather naively, as a mathematician, your role is to pick the relevant tools and arrange them in a meaningful sequence dictated by logic. One example of a surprising moment is when one tries to prove a theorem in algebra and borrows a technique from geometry, then restructures it to make it useful and relevant in algebra.

Borrowing techniques from other disciplines is another kind of abracadabra. Many mathematicians borrow techniques from physics, including quantum gravity, to prove theorems in geometry or topology by eliminating all physical notions and boiling them down to their essential mathematical arguments. It is surprising because not many can spot a complicated argument in physics, fully understand it, see through it, cut away the extra branches, and find the essence.

These tricks elicit a moment of awe because, essentially, they go beyond expectations. It needs an ‘eye’. But not every eye can capture that; experience can play a role, but it’s not enough. For that, I argue, one needs ‘vitality’, as in poetry.

Vitality in mathematical structure doesn’t die or become isolated. It moves, excites, and creates – it’s alive

Day-Lewis proposed that the object is not merely poetic per se; the object becomes poetic because of the poet. Good poets have high vitality; they live in the present, or, as the literature scholar John Livingston Lowes said, they don’t ‘ensconce themselves like hermit-crabs, generation after generation, in the cast-off shells of their predecessors’. They observe with a fresh eye to pin down an original thought. In fact, the originality of the image they see is directly linked to their vitality and to how connected they are to the present moment. The object becomes distinctly poetic as a result of their presence in the moment.

Mathematical objects, techniques and all the lemmas are the same. Sometimes, the experienced technical mathematician even knows them by heart and teaches them every year. They generously introduce open problems to novice mathematicians and patiently share every single detail about their progress. However, it takes a fresh PhD student just out of college to see the original image or a proof from THE BOOK. That, I argue, is because of their high vitality. This further results in vitality of the mathematical structure; the structure doesn’t die or become isolated. It moves, excites, and creates – in essence, it’s alive.

To conclude, my own definition of beauty in mathematics would be as follows:

A simple mathematical structure that surprises even the most experienced mathematicians and transfers a sense of vitality.

But is an AI-assisted proof simple or surprising? How do we define vitality in a machine? On these questions, the jury is out. Myself, I am torn. Maybe models just need more training to match our creativity. But I also wonder whether our limbic system is required. Can we write proofs without emotional kicks? I am also unsure if perfectly efficient brains can come up with novel revolutionary ideas.

Ultimately, this debate is about more than aesthetics; it is closely tied to the development of AI-assisted mathematics. If AI models can produce novel mathematical structures, how should we direct them? Is it a search for beautiful or truthful structures? A question that possibly guides the years to come.

Some mathematicians say they prefer the ‘truth’ and only the ‘truth’. However, my recent discussions with mathematicians showed me that most immediately recognise, enjoy and even wholeheartedly smile at a beautiful piece of maths. In fact, they spend their whole lives in search of one.

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