Breakthroughs in pure mathematics may take decades, and artificial intelligence is trying to speed up the process.

"Can artificial intelligence accelerate the pace of mathematical discovery?"

AI Faces Obvious Dilemmas in Mathematics

Artificial intelligence can write a poem in the style of Walt Whitman, offer dating advice, and suggest the best way to cook artichokes. But in mathematics, large language models like OpenAI's popular ChatGPT sometimes encounter basic problems. Some people believe this is an inherent limitation of the technology, especially when it comes to complex reasoning.

A new program by the U.S. Defense Advanced Research Projects Agency (DARPA) aims to address this shortcoming by recruiting researchers to find ways to conduct high - level mathematical research with AI "co - authors". The goal of this new grant program is to accelerate breakthroughs in mathematics, advance the pace of pure mathematics (as opposed to applied mathematics), and turn AI into the best mathematician.

Patrick Shafto, a mathematician and computer scientist at Rutgers University, said, "Mathematics is a great test - bed. It's a key pain point for current AI systems." Shafto is now a program manager at DARPA's Information Innovation Office I20. "If we overcome this, it will unleash more powerful AI," he added. "This has huge potential benefits for the mathematics community and society as a whole."

Put AI in the Forefront of Mathematical Testing Grounds

Dr. Shafto spoke in his office at DARPA headquarters, an anonymous building in northern Virginia. The blue - glass exterior gives few signs that it is one of the most unusual agencies in the federal government.

"By improving mathematics, we also learn how AI can work better," said Alondra Nelson, who served as a senior science advisor in President Joseph R. Biden Jr.'s administration and is a faculty member at the Institute for Advanced Study in Princeton, New Jersey. "So, I think it's a virtuous cycle of understanding." She suggested that in the future, AI proficient in mathematics could strengthen cryptography and aid space exploration.

After World War II, DARPA began competing with the Soviet Union in the space race. DARPA's most famous achievement was promoting research on ARPANET, the predecessor of the Internet we use today. The agency's small gift shop, which is not open to the public, sells replicas of cocktail napkins on which someone sketched the basic state of the computer network in 1969. DARPA later funded research that led to the development of drones and Apple's digital assistant Siri. But it was also responsible for developing Agent Orange, a powerful defoliant used for destructive purposes in the Vietnam War.

Andrew Granville, a mathematician at the University of Montreal, said of DARPA's mathematics program, "I'm sure it's not 100% pure (mathematics)," although he emphasized that he was only speculating about the final outcome. After all, DARPA is part of the Pentagon, even though it has historically operated with an enviable degree of independence. The U.S. military is rapidly integrating AI into its operations to avoid losing to potential competitors.

Meanwhile, Dr. Granville commended the effort as the Trump administration was cutting scientific research funding. "We're in a disaster period for U.S. science," Dr. Granville said. "I'm glad DARPA can funnel funds into academia."

One recent afternoon, Dr. Shafto, a 49 - year - old surfer and skateboarder, sat in a sparsely furnished conference room. He imagined that AI's ability to solve multi - step problems is similar to its ability to extract meaning from large amounts of text, which is achieved through probability theory.

Despite the unusually bad weather, Dr. Shafto wore a blue - and - white Hawaiian - style shirt, white flannel trousers, and sandals, with a Trilby hat on the table in front of him. Overall, his vibe was clearly closer to Santa Cruz than Capitol Hill, which largely aligns with DARPA's traditional disregard for the slow bureaucratic pace in the nation's capital. (The agency sets priorities and funds outside of scientists but does not conduct research itself; scholars like Dr. Shafto serve as program managers for an average of four years.)

The 'Crown' of Pure Mathematics

"There are great mathematicians working on old problems," Dr. Shafto said. "That's not something I'm particularly interested in." Instead, he hopes to save time by using AI, thereby speeding up the training process.

"Mathematical problems sometimes take decades or centuries to solve," he said in a recent speech at DARPA headquarters about the Mathematics Exponentialization project, which will accept applications in mid - July. Then he shared a slide showing that in terms of the number of published papers, mathematics had stagnated in the last century while life sciences and technological sciences had exploded. If that wasn't clear, the slide's title was: "Math is sloooowwww…".

The pure mathematics that Dr. Shafto wants to accelerate is often "slow - moving" because, unlike applied mathematics, it does not seek numerical solutions to specific problems. Instead, pure mathematics is an exciting field for visionary theorists who can boldly observe how the world works, and these observations are soon scrutinized (and sometimes torn apart) by their peers.

"Proof is king," Dr. Granville said.

Mathematical proofs consist of multiple building blocks called lemmas, and small theorems are used to prove large theorems. Bryna R. Kra, a mathematician at Northwestern University, admitted that whether each rickety tower of lemmas can maintain its integrity under strict scrutiny is what makes pure mathematics a "long and arduous process". "All mathematics is built on previous mathematics, so if you don't know how to prove the old stuff, you can't really prove the new stuff," she said. As a research mathematician, the current practice is to go through every step and prove every detail.

Lean, a software - based proof assistant, can speed up this process, but Dr. Granville said it "is annoying because it has its own protocols and language" and requires programming expertise. "We need a better way to communicate," he added.

The Risk of AI's Unexplainability

Can AI save the world? According to Dr. Shafto, that's the hope. An AI model that can reliably check proofs will save a lot of time and allow mathematicians to be more creative. "The constancy of mathematics coincides with the fact that we practice it more or less the same way: people still stand in front of the blackboard," Dr. Shafto said. "It's hard not to connect the two and say, 'Well, you know, maybe if we had better tools, it would change the progress.'"

Dr. Shafto and others believe that non - mathematicians will also benefit. Large language models like ChatGPT can search the digital human knowledge base and generate a semi - convincing college essay on literary history. But thinking through the many complex steps of a mathematical problem remains elusive.

Jordan S. Ellenberg, a mathematician at the University of Wisconsin - Madison and a member of the team applying for the Mathematics Exponentialization grant, said, "I think we'll learn a lot about the capabilities of various AI protocols from how they generate interesting material. We don't have an intuition about which problems are difficult and which are easy. We need to understand this."

A more disturbing fact about AI is that we don't fully understand how it works. Dario Amodei, the CEO of AI company Anthropic, wrote in a recent article, "This lack of understanding is unprecedented in the history of technology." Dr. Ellenberg noted, somewhat understatedly, that electricity was widely used before its properties were fully understood. However, since some AI experts worry that AI could destroy the world, any clarification of its operation is welcome.

Dr. Nelson, a former White House advisor, admitted that people's concerns about the rapid integration of AI into all areas of society are "reasonable". She believes DARPA is better suited to handle this case. "If you ask a question about Shakespeare and your chatbot hallucinates, there's a higher bar to meet," she said.

"The stakes are much higher."

This article is from the WeChat public account "Edu Guide", author: Alexander. Republished by 36Kr with permission.