When researchers at Microsoft released a list of the 40 jobs most likely to be affected by generative artificial intelligence (gen AI), few outsiders would have expected to see “mathematician” among them. Yet according to speakers at this year’s Heidelberg Laureate Forum (HLF), which connects early-career researchers with distinguished figures in mathematics and computer science, computers are already taking over many tasks formerly performed by human mathematicians – and the humans have mixed feelings about it.
One of those expressing disquiet is Yang-Hui He, a mathematical physicist at the London Institute for Mathematical Sciences. In general, He is extremely keen on AI. He’s written a textbook about the use of AI in mathematics, and he told the audience at an HLF panel discussion that he’s been peddling machine-learning techniques to his mathematical physics colleagues since 2017.
More recently, though, He has developed concerns about gen AI specifically. “It is doing mathematics so well without any understanding of mathematics,” he said, a note of wonder creeping into his voice. Then, more plaintively, he added, “Where is our place?”
AI advantages
Some of the things that make today’s gen AI so good at mathematics are the same as the ones that made Google’s DeepMind so good at the game of Go. As the theoretical computer scientist Sanjeev Arora pointed out in his HLF talk, “The reason it’s better than humans is that it’s basically tireless.” Put another way, if the 20th-century mathematician Alfréd Rényi once described his colleagues as “machines for turning coffee into theorems”, one advantage of 21st-century AI is that it does away with the coffee.
Arora, however, sees even greater benefits. In his view, AI’s ability to use feedback to improve its own performance – a technique known as reinforcement learning – is particularly well-suited to mathematics.
In the standard version of reinforcement learning, Arora explains, the AI model is given a large bank of questions, asked to generate many solutions and told to use the most correct ones (as labelled by humans) to refine its model. But because mathematics is so formalized, with answers that are so verifiably true or false, Arora thinks it will soon be possible to replace human correctness checkers with AI “proof assistants”. Indeed, he’s developing one such assistant himself, called Lean, with his colleagues at Princeton University in the US.
Humans in the loop?
But why stop there? Why not use AI to generate mathematical questions as well as producing and checking their solutions? Indeed, why not get it to write a paper, peer review it and publish it for its fellow AI mathematicians – which are, presumably, busy combing the literature for information to help them define new questions?
Arora clearly thinks that’s where things are heading, and many of his colleagues seem to agree, at least in part. His fellow HLF panellist Javier Gómez-Serrano, a mathematician at Brown University in the US, noted that AI is already generating results in a day or two that would previously have taken a human mathematician months. “Progress has been quite quick,” he said.
The panel’s final member, Maia Fraser of the University of Ottawa, Canada, likewise paid tribute to the “incredible things that are possible with AI now”. But Fraser, who works on mathematical problems related to neuroscience, also sounded a note of caution. “My concern is the speed of the changes,” she told the HLF audience.
The risk, Fraser continued, is that some of these changes may end up happening by default, without first considering whether humans want or need them. While we can’t un-invent AI, “we do have agency” over what we want, she said.
So, do we want a world in which AI mathematicians take humans “out of the loop” entirely? For He, the benefits may outweigh the disadvantages. “I really want to see a proof of the Riemann hypothesis,” he said, to ripples of laughter. If that means that human mathematicians “become priests to oracles”, He added, so be it.
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