In pure mathematics, very occasionally, breakthroughs arrive like bolts from the blue – the result of such inspired feats of reasoning and creativity that they seem to push the very bounds of intelligence. In 2016, for instance, mathematician , which has to do with finding the largest pattern of points in space where no three points form a straight line. The proof “has a magic quality that leaves one wondering how on Earth anybody thought of it”, he wrote.
You might think that such feats are unique to humans. But you might be wrong. Because last year, artificial intelligence company Google DeepMind announced that its AI had discovered a better solution to the cap set problem than any human had. And that was just the latest demonstration of AI’s growing mathematical prowess. Having long struggled with this kind of sophisticated reasoning, today’s AIs are proving themselves remarkably capable – solving complex geometry problems, assisting with proofs and generating fresh avenues of attack for long-standing problems.
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All of which has prompted mathematicians to ask if their field is entering a new era. But it has also emboldened some computer scientists to suggest we are pushing the bounds of machine intelligence, edging ever closer to AI capable of genuinely human-like reasoning – and maybe even artificial general intelligence, AI that can perform as well as or better than humans on a wide range of tasks. “Mathematics is the language of reasoning,” says at DeepMind. “If models can learn to speak it fluently, we will have created a very worthy intellectual partner.”
To understand the significance of AI taking on complex maths, you need to understand what human mathematicians do. Pure maths, as opposed to applied maths, is done with no practical purposes in mind. “Fundamentally, mathematicians are trying to understand,” says at the University of Wisconsin–Madison. They aim to find fundamental relationships and principles by studying abstract objects and concepts, such as numbers, algebra and geometry.
In practical terms, this involves a series of steps. You first define your terms. Then you combine these definitions into a mathematical statement, or a conjecture, that captures how these definitions relate to each other. And finally, you convert your conjecture to a theorem by writing a proof, demonstrating not only that the statement is logically true and valid for many scenarios, but also, hopefully, showing why it is true.
Mathematical reasoning
Pure maths, then, requires sophisticated reasoning, intuition and creativity. “Reasoning is quintessential to the mathematical process,” says at the University of Sydney in Australia. “In fact, it is difficult to find a discipline of human thought that exemplifies reasoning more.” That explains why mathematical reasoning has long been prized as something special, something beyond the reach of even the most powerful computers. “A machine can take hold of the bare fact, but the soul of the fact will always escape it,” wrote mathematician Henri Poincaré in 1908.
Mathematicians have used computers for decades, of course, but only for brute force calculations. When it comes to AI, even the wildly successful deep-learning neural networks – systems modelled on the way brains work – that have driven recent advances in the field have been unable to muster much in the way of mathematical reasoning. These days, however, there are signs that the latest AIs might be changing that.
Consider the Birch and Swinnerton-Dyer conjecture, a big open question in number theory. It concerns elliptic curves, defined by a certain kind of cubic equation in two variables, and in particular finding when these curves contain whole numbers. This conjecture is one of the seven Millennium Prize Problems selected at the turn of the century by the Clay Mathematics Institute in the US. Each is considered so challenging that anyone who can come up with a proof would bag a $1 million prize.
In 2019, some 70 years after this conjecture was put forward, at the London Institute for Mathematical Sciences and his colleagues wanted to see if AI could produce fresh insights. They directed a neural network to pore over a database containing millions of elliptic curves in search of a relationship between two mysterious properties – their rank, which is a characteristic of the curve, and a series of numbers called an L-function, which describe these curves in an alternate way.
The AI didn’t disappoint, finding that numbers in the L-function could be used to precisely predict the rank. After analysing the AI’s predictions in more detail, He and his colleagues found a striking statistical pattern that, when plotted on a graph, looked a lot like the fluid shapes produced by flocking starlings, known as murmurations. This work attracted more researchers, who have since found an equation to describe the pattern and found that it shows up in other important mathematical functions. “The AI was able to tell us to try something we wouldn’t have tried if we just used our own intuition,” says He.
It was a similar story a year later, when a collaboration between mathematicians and DeepMind used a machine-learning algorithm to rifle through large databases of objects from two more branches of maths: knot theory and representation theory. Again, the algorithm found potentially interesting relationships between certain aspects in the databases and researchers did further analysis to find genuinely new conjectures.
“Mathematicians were tremendously sceptical about whether machine learning would have anything useful to say about actual mathematics,” says Davies. Clearly, though, when it comes to spotting patterns in complex datasets, AI can perform tasks that human mathematicians can’t, even if its workings are sometimes a bit opaque. Working with these systems is like having a collaborator who can’t communicate well, says Williamson, who led the representation theory part of the project. “I always had the sense when working with these things that it somehow knows the answer, but it can’t tell me why.”
But there are reasons to think AIs can be more than taciturn pattern spotters. In 2022, OpenAI launched ChatGPT, a chatbot based on a form of AI known as a large language model (LLM). These offer fluent, human-like responses to all manner of prompts, having learned the patterns of language by ingesting vast swathes of text. Unlike the rest of the world, mathematicians were underwhelmed. But some researchers wondered whether ChatGPT’s underlying architecture, a type of neural network called a transformer, might be made into a more mathematically literate tool. The problem is that transformers, for all their capacity to generate text, are notoriously useless at filtering out wrong answers or spotting their own mistakes. Or, as Williamson puts it, “they don’t have a bullshit filter”.
So when DeepMind researchers built FunSearch, the system that produced the best-yet solution to the cap set problem, they created an LLM to write solutions to maths problems in the form of computer programs and combined it with a system that ranks the programs by performance. Those that work best are then fed back to the LLM, which iterates improved versions until it discovers something new. “It worked a lot better than I thought it would,” says Ellenberg, who worked with DeepMind to develop the system and collaborated with it to draw out fresh insights.
A different DeepMind team has since repeated the trick. This time, a similar set-up called AlphaGeometry tackled complex geometry problems from the International Mathematical Olympiad (IMO), a competition for the world’s brightest high school students. The IMO requires enormous mathematical creativity and AI systems have, historically, performed poorly when trying to answer its questions. But AlphaGeometry’s transformer model, which had been trained on made-up geometry problems, combined with a logical checking system, performed almost as well as the best humans.
Artificial understanding
Many mathematicians reckon this kind of combination might yield further treasures. Some even suggest it might be the first hint of mathematical creativity, too. “It could be possible that this is what creativity is,” says Williamson. “That a mathematician, like a poet or a musician or a novelist, just has a very good generational capacity and a very discerning evaluator.”
But these latest breakthroughs also suggest a more tantalising possibility: if the generative part of a system was trained on a vast corpus of research-level maths, rather than the school-level problems of AlphaGeometry, it could plausibly begin to find proofs for existing conjectures and suggest entirely novel proofs and conjectures without the input of a human. Arguably, that would amount to something akin to human-level reasoning and understanding – to grasping the soul of the fact, as Poincaré put it.
The problem is that the vast majority of cutting-edge maths can’t be read by a computer. The process of making it computer-readable, called formalisation, is tricky and time-consuming, and many mathematicians prefer to spend their time on the maths itself.
But formalisation is attracting more and more attention, not least because computer-assisted and computer-checked proofs are increasingly important in modern mathematics. Unlike in most other scientific disciplines, which test hypotheses through experiment and observation, mathematical knowledge is created through proofs. “Proofs are the centre of mathematics,” says at Johns Hopkins University in Maryland. “It is kind of what mathematics really is as a discipline.”
When Andrew Wiles famously proved Fermat’s last theorem in 1993, for example, he had to both discover and weave together cutting-edge results in number theory and algebraic geometry, which then spurred further research. But modern proofs, which can run to hundreds of pages, have also increasingly become a thorn in the side of mathematicians. Shinichi Mochizuki’s dense and impenetrable 500-page proof of the abc conjecture, another number theory problem, is yet to be verified despite being released in 2012.
Machines can help. Not only can they quickly check whether a proof is correct, they can also help mathematicians write their proofs. Until recently, the only major formalised proofs have trailed far behind current research and aren’t particularly useful for AI. But that is beginning to change. The idea is that if we can formalise enough proofs so AIs can access them, we could train these systems such that they can themselves generate conjectures and proofs more quickly, and maybe even more effectively, than we can. That way, AIs could learn to reason via mathematical thinking.
In 2022, Peter Scholze at the University of Bonn in Germany, a recipient of the Fields medal, the highest honour in mathematics, joined forces with his colleagues to formalise a new piece of his research using a computer language called LEAN. The project, called the , was completed in around two years and involved a new computer language invented by Scholze to describe topology, which concerns surfaces like doughnuts and spheres, using something called condensed sets.
Since then, other leading mathematicians have formalised their new research in LEAN. Last year, they and . The time gap between written and formalised proofs has shrunk from years to weeks.
Artificial general intelligence
That could be significant in terms of what AIs can do, says Kevin Buzzard at Imperial College London. “Once [AIs] start getting some ideas as to how mathematical objects or concepts work, because they’re reading our math libraries and getting the hang of how humans are using them, then you can imagine that maybe they’ll be able to prove things. But now there’s the question, can they invent new concepts?” In other words, will they be capable of producing new mathematical insights – those rare bolts from the blue – without human input?
For many mathematicians, the answer is a flat no, either that or they say such a prospect is at least decades away. The understanding and reasoning that they rely on for insights is uniquely human, and nothing that AI has done to date indicates that this has changed. “We’ve seen no evidence of this so far,” says Buzzard.
Even so, ever since they met for a Fields Medal symposium on the topic in 2022, many of the world’s leading mathematicians have been engaged in a debate regarding the extent to which automation will change the nature of their work, and how they might have to adapt. Some are more concerned than others that AIs present an existential threat to their profession, with much seemingly dependent on the extent to which AIs can ever decide on what mathematical ideas are interesting.
What AIs that can tackle advanced maths mean for the progress of AI as a whole, on the other hand, is a very different question – and one that may have implications for us all. Most observers say this is still many years away. But there is a select group of believers, buoyed by the rapid advance in LLM capabilities and what they have seen of their mathematical achievements so far, that thinks human-level mathematical reasoning – and by extension a more general form of AI – might be much closer.
The logic is clear enough: if mathematics is the highest form of human reasoning, then an AI capable of doing it as well as the best human mathematicians, or even better, would represent a significant stride toward artificial general intelligence (AGI), which is usually taken to mean a human-level intelligence that can handle all kinds of tasks. “The more we can push in that direction of intelligence… the more we’re moving towards AGI,” says Davies, even if he is quick to point out that true AGI would require a broader range of skills than reasoning alone.
, a computer scientist at the recently formed company xAI who has worked on using AI to do maths and automatic formalisation, is more bullish. He believes that we could have a superhuman AI mathematician by 2026. “Now, with new AI methods, they are basically artificial intuition machines,” he says. “They don’t just replace human intuition but exceed it by a large margin.”
If Szegedy turns out to be right – and it remains a big if – then machine mathematicians might be taking us further down the line to AGI than many would care to admit. And even if not, the challenges presented by high-level maths are clearly pushing AI to new heights.
“Mathematics is extraordinarily capable of describing many aspects of our universe,” says Williamson. “Suppose one had a system generally capable of answering difficult mathematics questions. Then such a system should also be generally capable of answering difficult questions about our world.”
Alex Wilkins is a news reporter at New Scientist
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