Can AI quicken the pace of math discovery?
Artificial intelligence can write a poem in the style of Walt Whitman, provide dating advice and suggest the best way to cook an artichoke. But when it comes to mathematics, large language models like OpenAI's immensely popular ChatGPT have sometimes stumbled over basic problems. Some see this as an inherent limitation of the technology, especially when it comes to complex reasoning.
A new initiative from the Defense Advanced Research Projects Agency seeks to account for that shortfall by enlisting researchers in finding ways to conduct high-level mathematics research with an AI 'co-author.' The goal of the new grant-making program, Exponentiating Mathematics, is to speed up the pace of progress in pure (as opposed to applied) math — and, in doing so, to turn AI into a superlative mathematician.
'Mathematics is this great test bed for what is right now the key pain point for AI systems,' said Patrick Shafto, a Rutgers University mathematician and computer scientist who now serves as a program manager in DARPA's information innovation office, known as I20. 'So if we overcome that, potentially, it would unleash much more powerful AI.' He added, 'There's huge potential benefit to the community of mathematicians and to society at large.'
Shafto spoke from his office at DARPA's headquarters, an anonymous building in northern Virginia whose facade of bluish glass gives little indication that it houses one of the most unusual agencies in the federal government. Inside the building's airy lobby, visitors surrender their cellphones. Near a bank of chairs, a glass display shows a prosthetic arm that can be controlled by the wearer's brain signals.
'By improving mathematics, we're also understanding how AI works better,' said Alondra Nelson, who served as a top science adviser in President Joe Biden's administration and is a faculty member at the Institute for Advanced Study in Princeton, New Jersey. 'So I think it's kind of a virtuous cycle of understanding.' She suggested that, down the road, math-adept AI could enhance cryptography and aid in space exploration.
Started after World War II to compete with the Soviet Union in the space race, DARPA is most famous for fostering the research that led to the creation of ARPANET, the precursor to the internet we use today. At the agency's small gift store, which is not accessible to the public, one can buy replicas of a cocktail napkin on which someone sketched out the rudimentary state of computer networks in 1969. DARPA later funded the research that gave rise to drones and Apple's digital assistant, Siri. But it is also responsible for the development of Agent Orange, the potent defoliant used to devastating effect during the Vietnam War.
'I'm sure this isn't 100% innocent,' Andrew Granville, a mathematician at the University of Montreal, said of DARPA's math initiative, although he emphasized that he was only speculating about eventual outcomes. DARPA is, after all, part of the Pentagon, even if it has traditionally operated with enviable independence. The U.S. military is rapidly incorporating AI into its operations, with the aim of not losing out to China and its People's Liberation Army or to Russia, which has been testing out new technologies on the battlefield in Ukraine.
At the same time, Granville praised the endeavor, which comes as the Trump administration is cutting funding for scientific research. 'We are in disastrous times for U.S. science,' Granville said. 'I'm very pleased that DARPA is able to funnel money to academia.'
A surfer and skateboarder in his free time, Shafto, 49, sat in a sparse conference room one recent afternoon, imagining a future when AI would be as good at solving multistep problems as it is at trying to glean meaning from huge troves of texts, which it does through the use of probability theory.
Despite the unseasonably raw weather, Shafto seemed dressed for the beach in a blue-and-white Hawaiian-style shirt, white flannel trousers and sandals, with a trilby hat on the table before him. His vibe was, on the whole, decidedly closer to that of Santa Cruz than of Capitol Hill, largely in keeping with DARPA's traditional disregard for the capital's slow, bureaucratic pace. (The agency sets priorities and funds outside scientists but does not do research on its own; academics like Shafto spend an average of four years as program managers.)
'There are great mathematicians who work on age-old problems,' Shafto said. 'That's not the kind of thing that I'm particularly interested in.' Instead, he wanted the discipline to move more quickly by using AI to save time.
'Problems in mathematics take decades or centuries, sometimes, to solve,' he said in a recent presentation at DARPA's headquarters on the Exponentiating Mathematics project, which is accepting applications through mid-July. He then shared a slide showing that, in terms of the number of papers published, math had stagnated during the last century while life and technical sciences had exploded. In case the point wasn't clear, the slide's heading drove it home: 'Math is sloooowwww. …'
The kind of pure math Shafto wants to accelerate tends to be 'sloooowwww' because it is not seeking numerical solutions to concrete problems, the way applied mathematics does. Instead, pure math is the heady domain of visionary theoreticians who make audacious observations about how the world works, which are promptly scrutinized (and sometimes torn apart) by their peers.
'Proof is king,' Granville said.
Math proofs consist of multiple building blocks called lemmas, minor theorems employed to prove bigger ones. Whether each Jenga tower of lemmas can maintain integrity in the face of intense scrutiny is precisely what makes pure math such a 'long and laborious process,' acknowledged Bryna R. Kra, a mathematician at Northwestern University. 'All of math builds on previous math, so you can't really prove new things if you don't understand how to prove the old things,' she said. 'To be a research mathematician, the current practice is that you go through every step, you prove every single detail.'
Lean, a software-based proof assistant, can speed up the process, but Granville said it was 'annoying, because it has its own protocols and language,' requiring programming expertise. 'We need to have a much better way of communication,' he added.
Could artificial intelligence save the day? That's the hope, according to Shafto. An AI model that could reliably check proofs would save enormous amounts of time, freeing mathematicians to be more creative. 'The constancy of math coincides with the fact that we practice math more or less the same: still people standing at a chalkboard,' Shafto said. 'It's hard not to draw the correlation and say, 'Well, you know, maybe if we had better tools, that would change progress.''
AI would benefit, too, Shafto and others believe. Large language models like ChatGPT can scour the digitized storehouses of human knowledge to produce a half-convincing college essay on the Russian Revolution. But thinking through the many intricate steps of a mathematical problem remains elusive.
'I think we'll learn a lot about what the capabilities of various AI protocols are from how well we can get them to generate material that's of interest,' said Jordan S. Ellenberg, a mathematician at the University of Wisconsin-Madison who is part of a team applying for an Exponentiating Mathematics grant. 'We have no intuition yet about which problems are going to be hard and which problems are easy. We need to learn that.'
One of the more disconcerting truths about artificial intelligence is that we do not entirely understand how it works. 'This lack of understanding is essentially unprecedented in the history of technology,' Dario Amodei, CEO of the artificial intelligence company Anthropic, wrote in a recent essay. Ellenberg somewhat downplayed that assertion, pointing out that electricity was widely used before its properties were fully understood. Then again, with some AI experts worrying that artificial intelligence could destroy the world, any clarity into its operations tends to be welcome.
Nelson, the former White House adviser, acknowledged 'legitimate' concerns about the rapid pace at which artificial intelligence is being integrated into seemingly every sector of society. All the more reason, she argued, to have DARPA on the case. 'There's a much higher benchmark that needs to be reached than whether or not your chatbot is hallucinating if you ask it a question about Shakespeare,' she said.
'The stakes are much higher.'
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