What it means when AI helps solve complex math problems

Host Megan McCarty Carino introduces OpenAI’s claim that its models solved one of mathematics’ Millennium Prize problems and later released more than 700 papers covering another 400 difficult problems. Rutgers mathematician Alex Kontorovich calls the results a “two-sided gift”: a human discovery would have prompted celebration, but the AI papers still contain “absolute gems.” He is especially excited by a quasi-Riemann hypothesis result that used familiar tools in an unusually clever combination, explored every branch of possible ideas, and reached a result humans might otherwise have found in 20 or 100 years.

Megan asks what it means that some papers include machine verification but not human verification. Kontorovich says mathematics is supposed to be more than an oracle declaring a theorem true: its purpose is to explain why, through logical deduction from axioms and rules. Humans must read the roughly 200-page arguments, test their truth, and understand their significance. He compares AI’s output to calories that have been dumped on mathematicians but are not useful until people digest and incorporate them into human mathematical knowledge and practice.

Kontorovich rejects the idea that mathematics education is dead because AI can generate proofs. Although some theorems matter because of real-world applications, he says most are pursued because humans want to solve difficult problems simply because they are difficult—like climbing a high mountain or going to the Moon. Human experts must still decide whether an AI result is interesting or valuable; leaving that judgment to AI alone would be “a pretty sad world.” Since expertise is trained by other humans, he expects the research-mathematics ecosystem to remain and possibly grow.

Kontorovich says AI creates a crisis of meaning because a person’s ability to write a deep paper once strongly indicated that they understood the mathematics. Now someone can press a button to produce text that experts may verify as true without that same evidence of comprehension. He predicts greater emphasis on verbal communication, oral exams, discussions, and lectures as ways to test depth of understanding. Responding to concerns from the Association for Humans in Mathematics—including fears that automated mathematics undermines careful thinking—he says hiring and graduate training rules are still unsettled.

Asked what an ideal human-AI partnership would look like, Kontorovich notes that OpenAI tested 4,000 problems but reported progress on only 400, leaving 3,600 untouched. He says that makes claims that mathematics is over premature. His preferred steady state is one in which AI acts like a compass pointing toward promising directions while mathematicians investigate them themselves. He expects the current chaotic period to settle into a new equilibrium over the next few years, although he cannot yet predict exactly where the field will land.