Mathematicians’ Chalkboard Writing Shows When Inspiration Strikes

Scientists see ‘Eureka’ moments in mathematicians’ writings on the board

By Matthew Hutson edited by Sarah Lewin Frasier

If you want to know when mathematicians are about to make a breakthrough, you don’t need to look inside their heads. Simply observe their movements on a board.

“I’ve always been very intrigued by this tension between the abstract, conceptual nature of mathematics, on the one hand, and the physical nature of actual mathematical activity,” says Tyler Marghetis, a cognitive scientist at the University of California, Merced. He wondered if he could use the “handwork” of mathematics to deduce what was going on in someone’s mind. In a recent study in the Proceedings of the National Academy of Sciences USAMarghetis and his co-authors borrowed theoretical tools from other fields to show that it’s possible.

Complex systems sometimes suddenly change state. This can happen when metals become magnetic, when algae takes over a pond, or when a horse changes from walking to trotting. Often, a period of instability precedes the tipping point. Some neuroimaging suggests that such a shift also occurs in the process of insight: when the brain gets stuck in a rut, falters and then finds the right path. This study illustrates this process at work.

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The researchers recorded six mathematicians in front of boards as they each spent about 40 minutes working on two math tests and thinking aloud. Observers noted each time a solver shifted attention to other parts of the board by writing, erasing, or pointing to equations, diagrams, or other markings. In this way, the mathematician and the painting acted together as what cognitive scientists would call an extended, semi-observable mind. The researchers also recorded insightful exclamations (“I see!”). Analyzing the data, they found that where attention turned became much more unpredictable in the two minutes before a eureka moment. It’s unclear where this unpredictability comes from: either a bubbling idea led the solvers to connect the puzzle pieces across the board, or the solvers had become frustrated and decided to physically search for new connections, which triggered a solution. Maybe it was a mix 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 just wish it would help me understand how to have more information,” he adds with a laugh. He would like to see the study’s statistical approach combined with in-depth interviews “to build a rich corpus of what mathematicians were thinking at the time.”

Psychologist Shadab Tabatabaeian of Georgetown University, lead author of the paper, imagines an “interesting application” of their method: One day, computer interfaces that track mouse or eye movements might know when not to disturb someone on the verge of a breakthrough or when to pitch them a new idea.

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