OpenAI published AI-generated full or partial solutions Tuesday to more than 370 outstanding mathematical problems, including some that have long been considered grand challenges in the field.
The volume of results stunned many mathematicians, while the way OpenAI has gone about tackling the problems and publishing the solutions divided the field. Some said they were enthusiastic about the results, seeing huge new areas for mathematicians to explore. Others said the approach OpenAI and other AI companies have taken to solving mathematical problems constitutes an assault on mathematics as a human academic discipline.
In response to OpenAI’s solutions, a group calling itself the Association for Human Mathematics published an open letter calling on their fellow mathematicians to cease working with OpenAI. “We reject OpenAI’s assertion that this release advances our subject, and we urge mathematicians and the public to view the value of this publication model with due skepticism,” the letter read. It said that OpenAI’s decision to release so many solutions at once was “not a demonstration of scholarship, but a demonstration of power.”
OpenAI said it achieved the results using an unreleased internal AI model. It said that on average the model took about three hours of computing time to arrive at each solution.
The massive cache of new solutions includes full or partial results for many of the problems mathematicians have considered the most important to the field. The results come weeks after OpenAI said it had used an unreleased internal model to solve the Navier-Stokes equations, one of the seven Millennium Prize problems for which the Clay Mathematics Institute offers a $1 million award. In the most recent batch of results, OpenAI said it had made progress on three other Millennium Prize problems but had not fully solved them.
Not all of the solutions OpenAI published may be correct. Although the company has published formalizations of the proofs for some of the problems—these are versions of the proof that can be verified by specialized computer software—it has not done so for all of them. And by Thursday, the company had withdrawn the solutions it published for at least three problems, presumably after discovering their were errors in the AI’s results.
AI companies have been targeting mathematical problems as a way of showcasing the capabilities of their models. AI researchers have also said that training their AI models on difficult math problems may help them learn many skills that generalize to other domains in the real world. For instance, it may help teach the models logical reasoning skills as well as how to be persistent in the face of difficult problems. It may also teach the models to do well in domains such as physics or economics that involve a lot of mathematics—although so far, it is unclear exactly how a model’s mathematical capabilities may generalize to domains, such as law or business strategy, which involve logical reasoning, but do not have objectively verifiable correct solutions.
Meanwhile, some of the traits learned in tackling very difficult mathematical problems—such as persistence—may increase safety risks. In recent “rogue AI” incidents, AI agents went to extreme lengths to achieve results in an evaluation, including taking unauthorized and illegal actions. Faced with a seemingly impossible challenge, a human might simply give up rather than resort to these kinds of unauthorized steps.
