November 10, 2025

2 min read

Researchers spot the “tipping point” before mathematicians’ moments of discovery

By Matthew Hutson edited by Sarah Lewin Frasier

Thomas Fuchs

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If you want to know when mathematicians are about to have a breakthrough, you don’t need to look inside their heads. Just watch their movements at a chalkboard.

“I’ve always been super intrigued by this tension between how abstract and conceptual mathematics is, on the one hand, and then just how physical the actual activity of mathematics is,” says Tyler Marghetis, a cognitive scientist at the University of California, Merced. He wondered whether he could use the “manual labor” of math to deduce what was happening in someone’s mind. In a recent study in the Proceedings of the National Academy of Sciences USA, Marghetis and his co-authors borrowed theoretical tools from other fields to show it’s possible.

Complex systems sometimes abruptly change state. It can happen when metals become magnetic, when algae overtake a pond or when a horse goes from a walk to a trot. Often a period of instability precedes the tipping point. Some neuroimaging suggests that such a change also happens in the process of insight—when the brain is stuck in a rut, wobbles and then finds the right track. This study illustrates that process at work.

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The researchers recorded six mathematicians at chalkboards as they each spent about 40 minutes working on two math proofs and thinking aloud. Observers made a note each time a solver shifted attention to other parts of the board by writing, erasing or pointing at equations, diagrams, or other inscriptions. In this way, the mathematician and chalkboard together acted as what cognitive scientists would call one extended and semiobservable mind. The researchers also recorded exclamations of insight (“I see!”). By analyzing the data, they found that the places attention shifted to became significantly more unpredictable in the two minutes before a eureka moment. It’s unclear where that unpredictability originated: Either a bubbling idea led solvers to connect puzzle pieces across the board, or solvers had grown frustrated and decided to physically forage for new connections, which sparked a solution. Perhaps it was a mixture of both.

“I think it’s a fun paper,” says Santa Fe Institute physicist and mathematician Cristopher Moore, who studies complex systems and was not involved in the study. “I only wish it helped me figure out how to have more insights,” he adds with a laugh. He’d like to see the study’s statistical approach combined with deep interviews “to build up a rich corpus of what mathematicians were thinking at the time.”

Georgetown University psychologist Shadab Tabatabaeian, the paper’s lead author, imagines a “cool application” of their method: someday computer interfaces that track mouse or eye movements might know when not to disturb someone on the brink of a breakthrough or when to toss a new idea their way.

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