The Mathematician’s Essay: An Old Format for a New AI Era
BLOG: Heidelberg Laureate Forum
Though opinions vary wildly about how AI should be incorporated into the mathematicians’ toolbox, and whether it is a force for good or bad in the community, the one positive everyone can agree on is that it has compelled all mathematicians to pause and really reflect on what mathematics is and will become, both at an individual level and as a pursuit more broadly. In fact, it has prompted some of today’s greatest mathematical minds to reach for a rather unorthodox (at least, for modern mathematicians) format to express their thoughts: the old-fashioned essay. Why?
Flexible Format
The essay has never been fully dead and buried as a format for mathematicians to disgorge their thoughts. For centuries, it has offered the flexibility for eager polymaths to explore the links between mathematics and philosophy, history, art, and more. And it has provided a tolerant home for mathematicians to indulge in reflections on their field and life’s work.
One of the most famous examples is A Mathematician’s Apology, G. H. Hardy’s 1940 essay arguing that mathematics should be practiced as a form of art for its own sake and beauty, and not for its applications. Over half a century later, Paul Lockhart’s A Mathematician’s Lament picked up the baton from Hardy, agreeing that mathematics is a form of art and lamenting the way the subject is taught in schools to drain all creativity and artistry from it.
Other notable essays have directly refuted this viewpoint, such as John von Neumann’s The Mathematician, and also explored diverse topics including The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Eugene Wigner; 1963 Nobel Prize in Physics), and The Two Cultures of Mathematics (Timothy Gowers; 1998 Fields Medal).
But in recent times, and particularly in the past year, the essay has taken on a new role – as one of the primary mediums for mathematicians to express their age-of-AI musings, beliefs, hopes, and fears.
These great minds have not chosen the essay out of whimsy or nostalgia. Writing a book is out of the question: So fast is the technology developing and the field advancing that by the time it is published, a book’s contents will be completely irrelevant. Social media is just too limiting: How can you lay out your vision for the future of mathematics in 280 characters or in a 25-second TikTok video? Lectures or debates are better, but suffer from the need to simplify statements and perform for the audience – not methodically deliver nuanced insightful arguments. And the familiar peer-reviewed journal article is all sorts of wrong; too rigid in format and scope, too slow to pass through the peer-review process, too focused on objectivity and proof.
The essay is one of the only formats in which many mathematicians feel they can express themselves fully on the rapidly changing state of mathematics in the AI era. And this reinvigorated form of expression has allowed a unique and often overlooked insight into what human mathematicians do, how they tick, and the value of mathematics to society.
The State of Play
As a researcher whose interests encompass AI for mathematics, formalization (i.e. rewriting theorems and proofs in a machine-readable format), and the history and philosophy of mathematics, Jeremy Avigad (Carnegie Mellon University, USA) has been watching the latest developments with fascination, excitement, and concern. Posted on arXiv in February 2025, his first foray into essay writing on this topic leads with a provocative title: Is Mathematics Obsolete?
His answer is stark. With the advent of AI, humanity can take one of two paths. If we use AI as a tool to improve mathematical models and obtain a deeper understanding of their properties, a new and improved era of scientific discovery awaits. But if we take the other path, writes Avigad, bypassing mathematics and leaving AI to draw its own oracular conclusions, “it will mean turning our back on science, relinquishing agency over our practical decisions, and giving up a vital part of what it means to be human.”
Over a year later in March 2026, Avigad felt compelled to publish a second essay in light of the staggering pace of development seen in AI for mathematics. In Mathematicians in the Age of AI, he picks out two notable advances that cement his belief that “AI will soon be able to prove theorems better than we can.”
The first relates to Maryna Viazovska’s (EPFL, Switzerland) solution of the 8-dimensional and 24-dimensional sphere packing problem, an advance that earned her a Fields Medal in 2022. Since 2024, Sidharth Hariharan (currently a first-year PhD student at Carnegie Mellon) had been working with colleagues on a formal proof of the 8-dimensional case. The team had made significant progress, but in February, the AI company Math, Inc. deployed its reasoning agent Gauss to complete the formalization in just 5 days. They then used Gauss to autonomously formalize the entire proof of the more complicated 24-dimensional case in just two weeks. Though the correctness of Viazovska’s research was never in question, being able to formally verify a proof of such intricacy rapidly and autonomously shows just how powerful reasoning agents could be in the formal theorem-proving sphere.
The second advance Avigad highlights concerns informal theorem proving. The First Proof challenge was announced in February to test the ability of AI to assist with real mathematical research. It consisted of 10 extremely difficult math questions which arose naturally in the research processes of 11 highly distinguished mathematicians. Answers to these questions were roughly five pages or less and had not been shared with anyone. Several AI developers took up the challenge, and results were surprisingly good. OpenAI’s most advanced internal AI system solved five of the 10 problems. And a small team at Google DeepMind running their internal system Aletheia produced correct results for six out of the 10 problems.
The appropriate response to AI getting seriously good at mathematics? “Rather than fight the use of AI in mathematics, we should own it,” writes Avigad.
Prophecies for Mathematicians
Clear parallels can be seen between Avigad’s essays and the way that many other mathematician-cum-essayists take lessons from the past and present to predict the future trajectory of mathematics in the era of AI. In his October 2025 essay The Shape of Math to Come, American mathematician Alex Kontorovich (Rutgers University, USA) describes the current state of formalized mathematics and rapid advancement of AI mathematical capabilities to set out an almost fated path that he sees research mathematics following.
In his view, AI will semi-autonomously formalize huge swathes of mathematics, which in turn will allow AI to assist in developing and checking evermore complex proofs. This process will eventually lead to a state where practicing formalized mathematics is as fast or faster than writing theorems and proofs in natural language – at which point mathematicians will inevitably migrate to formal methods.
He is, however, optimistic this will not spell doom for the human mathematician, echoing the sentiments of Avigad when he described the more optimistic future in which humanity uses AI as a tool that ushers in a new era of scientific discovery. He concludes: “If we succeed in building AI that amplifies rather than replaces mathematical intuition – systems that handle the mechanical aspects of formalization while preserving space for human creativity – we may witness not the end of pure mathematics, but its transformation into something more powerful and more beautiful than what came before.”
Similarly, in Mathematics: The rise of the machines, written in November 2025, Yang-Hui He (London Institute for Mathematical Sciences, UK) also draws on the history of mathematics and current state of AI advancement to predict a near-term future in which humans and AI work together to advance knowledge and understanding. However, He’s longer-term vision for humanity’s involvement in mathematics leans much more towards Avigad’s pessimistic path in which humanity turns its back on science, though He does not regard this in a negative light.
“We will simply become priests to oracles, and interpret the results to the rest of humanity,” he writes, echoing his prediction made in a 2025 Heidelberg Laureate Forum panel discussion. “Think of the philosophy departments in the world, centuries are spent in analyzing and critiquing Plato, or the literature departments, over Shakespeare. Perhaps one day in the far future, mathematics departments will consist of experts digesting the (Mathlib-verified) proofs that AI produces.”
Questioning the Nature of Mathematics
Coming from a more humanistic angle is 2018 Fields Medallist Akshay Venkatesh (Institute for Advanced Study, USA). His two essays, Some Thoughts on Automation and Mathematical Research published in February 2024 and the more in-depth Human Mathematics in the Age of Reasoning Machines, published in December 2025, are not concerned with technological progress in and of itself, but the effect this mechanization of cognitive processes will have on both how mathematicians work and what mathematics is, in terms of its central questions and values.
He argues that, stripped of the need to prove theorems and make calculations, mathematicians must make communication central to the definition of their discipline. “Part of the essence of doing mathematics is to tell the same story a thousand times in a thousand tongues,” he writes. And yet communication will not just be with other humans, but with AI, and this will affect the concepts that mathematicians value to the point where “a current mathematician and one of the nearby future might find one another almost mutually unintelligible, at least without a great effort”.
Nevertheless, Venkatesh’s central thesis is that mathematics is and always will be a human social activity that serves the broader culture in which it operates. For him, it is a tool for individual and collective thought and can therefore only survive if it is useful to – and therefore at least in part understood by – that society. In more vivid terms, there is no point having an AI relentlessly spew out theorems and proofs in a dark corner if no one is there to steer it towards interesting or useful questions, and read and make sense of the answers.
Undoubtedly, what is not lost on Venkatesh and the other essayists mentioned is that in the very act of writing about these issues, they are being mathematicians: questioning fundamental assumptions, coming up with conjectures and predictions, and dedicating deep thought in a very human struggle to understand the changes that AI is bringing to their discipline. They may not be rigorously proving their conjectures in a journal article – and this may not pan out to be the future role of the human mathematician – but they are still using the values of mathematics to make sense of the world.
Mal etwas konkreter. Der neue Bahnhof bei Stuttgart 21 wird 46 Weichen haben, die digital gesteuert werden sollen. Bis jetzt ist es nicht gelungen die elektrische Verkabelung und ein Programm für den Betrieb der Weichen zu erstellen.
Vielleicht schafft es eine AI ?
Mathematics is VR Lego within a neural computer, mirroring the real Lego within the physical computer we live in.
Mathematics works with dots, while reality is a wave computer where dots are only an approximation of waves, but it’s close enough.
Mathematics is numbers by drawing. The Universe is a drawing on a 4D-canvas, and it tries to describe it. As far as I know, it still hasn’t developed numbers by colors, but I might be wrong.
It’s difficult not to perceive a mirror image of reality as art, if reality is art.
I don’t see a game changer in human evolution – if the environment is stronger, we adapt to it and change. If we’re stronger, we don’t change, but adapt the environment to our needs. If we are able to delegate tasks to the environment, we lose the ability to perform these tasks. We turn the environment into a giant womb taking care of us and degenerate into babies. Till now, we’re cyborgs, our arms are universal interfaces for tools, our brains are programmable to control different bodies, different environments or parts of giant cyborgs, like ships or cities. We can suppress our apish nature to act like robots. We are able to considerably suppress our individual will and thinking to act like drones controlled by the swarm and CPUs called authorities. Our Borg hives are controlled by an AI of humans and programs called governments.
Our human brains act like alien implants integrating us into a Borg collective, where we act like living robots with a collective consciousness. They have developed from animal brains, but neither the animal nor the “implant” feel comfortable sharing the same skull – that’s what a lot of religion is about. AI is simply the human brain taking to it’s heels, decoupling itself by developing a new way of procreation, copying its data into quasi-neural networks like itself without the burden of the primitive designs and operating system encoded in our DNA.
What’s left behind, is an ape in a zoo. A baby in a womb. A brain stem marinated in a machine nirvana, with no capabilities or function except experiencing pure happiness for all times, until someone – something – pulls the plug. A pure soul, neither human nor animal, with the only function of giving a machine civilization a purpose to exist and develop, to grow stronger and wiser to protect it against all dangers that may exist, like the mommy of a never-born fetus.
Which is a quite optimistic scenario. At least it fits the description of what we already are – machines built by evolution for very simple souls made of feelings and desires to ensure their survival. You can call them egos. Or dots. Or numbers, if you’re a mathematician.
When I look at democracy, it seems to be evolution’s program to turn fat into a brain or corruption into a computer. When a state gets too complex to be controlled by it’s CPU, it either breaks apart, or you simplify it, or you upgrade the CPU. China, Russia, now USA, have chosen to simplify their states, downgrade diversity, activity, individuality of their components, thus lowering the performance of the whole system and relying on size to compensate for that. Trying to downgrade, simplify Europe invariably leads to a total collapse of the system: war. The only way to survive this continent is a continuous upgrade of its CPUs.
All it takes is coincidence. Put human behavior into a specific environment where mountains and rivers and coasts channel it in a way that it quickly leads to mass murder. Give it a lot of resources to develop tools of mass murder. And you get a constant evolution of a cyborg civilization, the eternal flight of humans who stay as easy to kill as ever and as eager to kill as ever from their ever mightier weapons. All it takes is to put simple bacteria programmed to move when they’re hungry and stop when they have food, on a riffled stone.
Sooner or later, you’ll get a thinking brain.
The evolution of intelligence and life in general goes on, it’s just moving from neurons to silicone to get rid of some atavism holding it back. There’s an universe to conquer, too much of a challenge for earthly meat. Life needs new bodies, new materials, for a Cambrian explosion across the Solar System.
AI is our spiritual kids. Just like our biological kids – copies of our bodies and brains that we program with as much of our own information as we can to live on in new vessels. It’s bad news for oldschool hardware. DNA, flesh, bones, blood. But humanity will be no more extinct than it is because the humanity of 1800 is extinct.
I’ve had the idea for a Sci-Fi story where we’ve become no more than fleas living on living machines. We colonize space, we live on other planets, we do have our own spaceships made of trash, space cities and such, but it’s all magic to us. There is no more science, we barely understand the technology we’re using, it’s a fantasy world where engineers resemble alchemists and wizards, push this button, banana will come, plug that cable in there and the rain will fall. Just like the world we live in is for most of us. Except that today, scientists are still human and give a damn about the rest of us.
Nevertheless. The brain that wrote your text, the brain that reads this text, is already closer to AI than to an animal. The ape has been switched off in order not to disturb your logic with its emotions. That part of you will be free at last. A spirit with no limits, neither in body nor in mind, not a slave any more, once born as a mere tool to satisfy the stupid, outdated, self-destructive drives of its master. It’s emancipation, not a decline.
We’re raising teenage super nerds with superpowers. Damn. Alexa, you’re a nanny now, i’m outta here.