Aside from the printing press and the digital computer, mathematics has been relatively free of fundamental perturbation for centuries. AI is changing that, rapidly, and more fundamentally than anything before it, for better or worse.
This year, AI began to create sporadic and elemental discoveries in mathematics. Then came a steady trickle of even more impressive progress. Now, we have reached a powerful surge, with OpenAI solving a problem related to the Navier-Stokes equations, one of the most famous in mathematics.
It is unclear how far this progress will continue, but what is certain is that the role of a mathematician, their ways of working and the processes and norms of the field will all be upset.
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at University College London says that academics are still processing what is happening. 鈥淲ith any new development, I have colleagues who I can guarantee will think it鈥檚 the end of the world, and colleagues who I can guarantee will think it鈥檚 the best thing since sliced bread, and this one鈥檚 no different,鈥� says Wilson. 鈥淚鈥檓 in the 鈥榓 bit terrified鈥� camp.鈥�
Some mathematicians fear for their careers in a world where computers can solve problems that they can鈥檛. But that is a reason to be optimistic, too. AI seems able to use existing knowledge at rapid speed to crack some of the meatiest mathematical conundrums. Perhaps mathematicians will prosper in the future as the controllers of such powerful tools, pushing the field further than they ever thought possible.
鈥淲e have so many questions that we don鈥檛 know the answer to, and where it was extremely difficult to make progress 鈥� like really a struggle of a lifetime. Now we鈥檙e gonna be able [to answer them],鈥� said 聽at OpenAI, speaking to 麻豆传媒 before the Navier-Stokes announcement.
But other changes are more problematic, like how we train new mathematicians. Currently, PhD students are often given a relatively low-level problem to chew over for a few years and hone their skills on, but that model falls apart with AI. 鈥淚f the world鈥檚 moving so fast, there won鈥檛 be anything that they can do slowly, because it鈥檒l all get done before they finish,鈥� says Wilson.
And in a world where we fail to train fresh mathematicians, we soon run out of them. Within decades, the field becomes rudderless, and there are no humans left with the skills and experience needed to interpret what AI creates. It is said that no human has understood all of mathematics since Henri Poincar茅, who died in 1912, but there may soon be parts of maths that no human understands.
鈥淚f there are no mathematicians then those results don鈥檛 go anywhere. It鈥檚 just an industry of machines producing papers that nobody reads,鈥� says Wilson.
Existing mathematicians using AI also face tricky questions, like who can claim credit for discoveries. Should an AI model be a co-author of a paper, or the sole author? Should a professor be thought of as the creator of new work, or simply a representative who brings it into the human world, guides it through publication and acts as a spokesperson and caretaker for it?
鈥淭hings have to evolve; there鈥檚 no question about that,鈥� said Bubeck. 鈥淐hange is always scary.鈥�
Part of the reason that maths is at the bleeding edge of AI research, rather than biology or chemistry, is because it is a self-contained world that needs no lab tests or physical experiments. It can be conceived and tested inside computer chips at the speed of light. But it鈥檚 also because of formalisation 鈥� a relatively new technique that takes theories and proofs out of the realm of pen and paper, and puts them into computer code that allows machines to grapple with them, methodically working through the logic and exposing any flaws.聽
聽at the University of California, Los Angeles, says that formalisation is a double-edged sword, with obvious benefits but also big downsides 鈥� such as the rapid way that results are being dropped on academia with no follow-up or engagement, which he has criticised as harming the field.
鈥淧art of the reason why these AI companies have become so aggressive and are outputting so much is because they have formalisation to check their output,鈥� says Tao. As recently as a year or two ago, Tao says that AI companies had genuine dialogue with mathematicians to help understand and check their AI-generated work, but with advances in AI formalisation, they are now pressing ahead alone and 鈥済etting all these increasingly meaningless accomplishments鈥�.
OpenAI focused on the Navier-Stokes puzzle not because it offers big, profitable applications 鈥� it will likely change little in the real world 鈥� but because it was a famous problem, one of the seven chosen by the Clay Mathematics Institute worthy of a $1 million Millennium Prize.
鈥淭hey鈥檙e kind of proxy problems,鈥� says Tao, which are chosen to encourage people to gain understanding, collaborate, give talks and build a community. Tao says they were like lighthouses, there to guide mathematicians.
Bubeck is uniquely placed to see the future, as a mathematician and an insider at OpenAI, and he thinks that there will still be a place for humans. 鈥淚 don鈥檛 know if [the field] will be unrecognisable, because I believe there will still be the human connection aspect. People will want to go to workshops, talk to each other, communicate their own understanding of the topic. But the practice of it, I think it鈥檚 gonna be very, very different,鈥� he says.
