The Mathematician’s Essay: An Old Format for a New AI Era
Benjamin Skuse
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.
The post The Mathematician’s Essay: An Old Format for a New AI Era originally appeared on the HLFF SciLogs blog.