Mathematicians’ New Best Friend?
Benjamin Skuse
When the 101 young mathematicians were making their way to the 12th Heidelberg Laureate Forum from across the world, most would have been excited by the potential for meeting new friends, new collaborators, hearing inspiring talks and potentially finding new research directions. Few would have expected to be considering whether mathematics was on the cusp of a transformative AI revolution after the first day. Yet, the message from Sanjeev Arora (ACM Prize in Computing – 2011), David Silver (ACM Prize in Computing – 2019) and Richard S. Sutton (ACM A.M. Turing Award – 2024) during their Spark Session talks was clear: They expect that AI will unshackle itself from its creators to transcend human knowledge and capabilities, and the first subject to feel the effects of this new world order will be mathematics.
From various discussions with young researchers during the coffee breaks, this message gave many of them a somewhat bitter aftertaste. Excitement that a superintelligent AI mathematician might be able to solve the Riemann hypothesis and other important unsolved problems in pure mathematics was tempered by more practical concerns. In five to 10 years’ time, will these young mathematicians still have a job? Will the generation after that? And if so, will humans still be conducting the part of mathematics that is fun; the creative, problem-solving side at the bleeding edge of current knowledge?
On the second day of the Forum, it was into this somewhat unsettled atmosphere that the Hot Topic session “The Machine-Learning Revolution in Mathematics and Science – Part 1: Mathematics” kicked off. Panellists Yang-Hui He (Fellow of the London Institute of Mathematical Science, UK), Javier Gómez-Serrano (Professor at Brown University, USA), Maia Fraser (Professor at the University of Ottawa, Canada) and Arora, were asked by moderator George Musser what excites and concerns them most about AI. Their answers mirrored the thoughts of young scientists and can be broadly summarised as: New vistas in mathematics will be opened up by AI, but the technology is developing too rapidly to understand what roles humans will play in the mathematics of the future.
Picking up the Pace
To these concerns, there was a wide array of responses from the eclectic expert panel. Gómez-Serrano is working at the coalface of AI development, and so has a clear view of the pace of progress. He was first exposed to the power of AI when he began applying neural networks to find out when and if Leonhard Euler’s 1757 equation to describe the motion of an ideal, incompressible fluid, ever breaks down and delivers nonsense.
More recently, Gómez-Serrano has been working with Terence Tao (Fields Medal – 2006) and Google DeepMind to develop AlphaEvolve. AlphaEvolve is a Gemini-powered evolutionary coding agent that substantially enhances capabilities of state-of-the-art large language models on highly challenging tasks such as tackling open scientific problems or optimising critical pieces of computational infrastructure. “It’s doing meaningful things in actual open problems at a much faster time scale than a trained human,” he said. “In the maths community, progress was measured in years or even in decades, and now we can start to measure things in months.”
Arora also has a well-resolved view of progress in the field. He received the ACM Prize in Computing 2011 for contributions to computational complexity, algorithms and optimisation that have helped reshape understanding of computation. In 2023, he became founding Director of Princeton Language and Intelligence, a new unit at Princeton University devoted to the study of large AI models and their applications, and currently he is deeply involved in developing Goedel-Prover, an open-source language model that generates automated formal proofs of mathematical problems.
Arora’s message was even more sobering than Gómez-Serrano’s: “The number one takeaway from the AI revolution of the last 10 years is that whatever we thought was hard for AI is often easy, and vice versa,” he exclaimed. “And my guess is that making good conjectures is the stuff AI will be good at; and that’s the creative part.”
Beyond the Turing Test
Although He sees himself as a “fanatic” in the use of AI and machine learning in pure mathematicians, he suggested AI is still far from the finished article, with time left before it starts benching human mathematicians. With ChatGPT passing the Turing test in 2022, He and colleagues came up with a new stricter test for AI-assisted conjecture formulation, which they named the Birch test, consisting of three components: A, I and N. A stands for automaticity, such that the conjecture raised by the AI cannot be influenced by humans during its process. I is for interpretability, so that humans can interpret the mathematics in a form they can digest. And N is for non-triviality, in the sense that the conjecture is interesting enough for humans to want to work on it. “In this very strict sense, nothing out of the hundreds of papers in the past decade has passed this Birch test,” he remarked. “There are only two instances where we got close.”
The first He mentioned was a 2021 DeepMind paper published in Nature in which a machine-learning-guided framework was used to find a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups. “That passed the A and I,” said He. “But it didn’t pass the N test because they were able to prove it.”
The other case was one of his own published in 2024: the murmuration conjecture. He and colleagues used machine learning on millions of elliptic curves (i.e. curves defined by equations of the form \(y^{2} = y^{2} + ax + b\), where \(a\) and \(b\) are constants) to predict their ranks (numbers indicating how many independent points are needed to generate all other points on the curve). Analysing and visualising the success of the method revealed patterns that resembled a murmuration – the ever-changing striking patterns that large groups of starlings make – with curves of different ranks flowing across the plot. “But that failed the A test,” said He. “Because we had to choose the algorithm to try to inch out that formula.”
Human Agency in AI Deployment
In contrast to the other panellists, Fraser felt that the replacement of humans in important tasks like forming conjectures is far from inevitable, and still in the hands of human mathematicians, at least for the time being. A mathematician researching machine learning, for the past five years Fraser has been running a training programme across four Canadian universities to support graduate students who are combining mathematics and machine learning, but also to raise their awareness about questions of social impact and ethics.
Then last year, Fraser was one of the guest editors for two special editions of the Bulletin of the American Mathematical Society called “Will machines change mathematics?”, whose call to action – “It is for us to determine how our subject should develop” – resonated with the community and forced many to start seriously thinking about the impact of AI on mathematics and their own work. She is also currently writing a book on AI safety and has spoken on this topic for the HLFF Blog in a previous post. “I would emphasise the need to just pause for a moment and try to see a big picture view,” she stated. “The concern is that without looking at the social impact of some of these things, stuff will just happen by default. But we have a lot of agency in terms of the kinds of things we want to invent and how we’re going to deploy them.”
Priests to Oracles?
Though the comments from panellists were enlightening, they did not quell the audience’s concerns about rapidly evolving AI posing risks to human mathematicians. Unsurprisingly, considering the pace of change, there was little consensus from the panel to the audience’s insightful questions and comments.
For instance, one young researcher asked whether the uptake of AI for research will lead to institutions putting less resources into recruiting and training people to do research work. “The way you do research will change, and people will get used to AI assistance,” said Gómez-Serrano. “But it may or may not affect hiring.” Fraser, in contrast, was certain that AI uptake would affect the job market, calling for institutions and society more generally to recognise the importance of jobs in mathematics as not only performing a function and a means to live, but also as a way to discover purpose in life through the community noticing what someone is doing and appreciating it: “We should acknowledge that and value it so that we can make sure that it’s cared for as we navigate into the future,” she said.
Several audience members asked questions about what could happen if AI gets to the stage where humans are completely out of the loop. Arora and He, in particular, seemed unfazed by this idea, suggesting human mathematics will morph into an activity akin to chess or studying English Literature. “Just as chess now is more popular than ever, even though machines have been much better for 25 years, humans like to do it, and so they will like to do math,” Arora claimed. “I don’t know how to predict how that will all work, but people will enjoy doing it more, because from an early age, people will have this amazing [AI] teacher, which very few people had growing up previously.”
“A colleague of mine from the English Department gives a good view on this,” began He in his response. “Imagine the extreme case where [mathematics] is all done completely by AI: Where do we come in, and what do we do? And my friend says: ‘Well, what do English professors do? We write commentary on Shakespeare,’ … and that’s still a very, very interesting thing to do. So in the distant future, where you have an automatically generated proof of the Riemann hypothesis, we can then write commentary on how we digest the proof and make sense of it and talk about it. In some sense, mathematicians become priests to oracles.”
Only Fraser made the point that AI only gets to the stage where humans are completely out of the loop if humans choose to take themselves out of the loop, re-emphasising that right now mathematicians have agency to plot the course of how AI is deployed in their profession, forming a template for other areas of society that will be affected later down the line. “One thing that young researchers in math can do is identify things that they value in being a mathematician, and care for those so that the path in which AI is integrated into math honours that,” she concluded. “Math is a very cohesive community; it’s going to be much harder in other areas of society, so it’s important that math sets an example for these societal aspects of AI.”
The post Mathematicians’ New Best Friend? originally appeared on the HLFF SciLogs blog.