How AI is reshaping discovery in maths and physics
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How AI is reshaping discovery in maths and physics
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Illustration: Ana Kova
Among mathematicians and theoretical physicists, artificial intelligence provokes a range of reactions. Some see it as irrelevant to their work; others fear it could encroach on the most creative, intellectually rewarding aspects of their fields. Yet, the truth that’s emerging, from the work our team is doing at the London Institute for Mathematical Sciences and elsewhere, is subtler.<br>Rather than displacing human creativity in mathematical sciences, AI is augmenting it. Software can now check proofs line by line and catch errors that would once have taken months of human scrutiny to find. It can search systematically for counterexamples — testing whether a conjecture truly holds or fails in an unexpected way. And it can propose intermediate steps in an argument, suggesting useful auxiliary results that help to bridge the gap between what is known and what still needs to be shown.<br>AI cracks 80-year-old mathematics challenge — researchers are astonished
In experimental fields, prototype ‘AI scientists’ are beginning to automate parts of the discovery cycle, but they remain constrained by the demands of the physical world: mixing reagents, culturing cells, waiting for reactions and contending with noise in the data. Mathematics and theoretical physics face many fewer bottlenecks. ‘Experiments’ are cheap, fast and digital, and mathematical data — from prime numbers to the properties of abstract structures, such as manifolds — are clean and abundant1.<br>Companies developing AI systems tailored to mathematical reasoning have reported steady progress in the past year. Aristotle, a system from software company Harmonic in Palo Alto, California, has helped to solve several problems posed by the prolific mathematician Paul Erdős — questions that are easy to state but notoriously hard to crack. Axiom Math, a start-up company in Palo Alto, has announced that its AI tool found solutions to many research-level problems that professional mathematicians had not yet solved. Meanwhile, models from technology firms OpenAI in San Francisco, California, and Google DeepMind in London have solved several challenges from the First Proof Project, a set of difficult mathematical problems that test whether AI systems can generate new and verifiable results.<br>Here, we give examples of progress in the past few years in this rapidly evolving area, outline the opportunities that AI presents to scientists and mathematicians in theoretical domains — and invite researchers to lean in to using AI in their work.<br>The research pipeline<br>In theoretical physics and maths, researchers weave together creative insight and rigorous logical reasoning to make discoveries — but this process is only partly understood, and there is no single explanation for how breakthroughs happen. For clarity — without putting forth a definitive model — we break the process into several overlapping phases: setting the agenda, formalizing ideas, proposing conjectures and solving and verifying results. This framework is imperfect, but it provides a useful way to assess where AI is already contributing, where challenges lie and how they might be addressed.<br>Setting the agenda. One of the most distinctly human acts in research is deciding which questions are worth asking in the first place. These might arise from outside the field — through real-world problems or contact with neighbouring disciplines — or from within it, in that theories evolve according to their own internal logic and aesthetic standards2,3. These sources are intertwined: concrete problems can generate new concepts, and abstract theory can reshape and deepen the original question.<br>‘It is incredible’: How AI is transforming mathematics
Today’s AI systems have only limited access to this broader context. As a result, they lack intuition and ‘taste’: a sense of where questions come from, what makes them timely and how they fit into a field’s evolving structure. For instance, physicist Albert Einstein developed his special theory of relativity after noticing a contradiction in how light waves were treated in classical mechanics and in Maxwell’s equations, which describe the interplay of electricity and magnetism.<br>One promising but under-explored direction is to build AI systems that help to sort and prioritize potential problems using criteria selected by researchers. For example, AI could follow those criteria when scanning large mathematical databases,...