The eureka moment in mathematics feels like it comes from nowhere. A new study suggests it does not. In the minutes before mathematicians announce a sudden insight, the way they move between the sketches and equations on their blackboard becomes measurably less predictable.
The research, “An information-theoretic foreshadowing of mathematicians’ sudden insights”, was published online in the Proceedings of the National Academy of Sciences on August 18, 2025. The authors are Shadab Tabatabaeian and Tyler Marghetis of the University of California, Merced, Artemisia O’bi of Indiana University, and David Landy of Indiana University and Netflix.
Filming proofs in the wild
The team video-recorded mathematicians working alone at blackboards in their own offices and seminar rooms. They tackled problems from the Putnam competition, a notoriously hard contest for undergraduates in which the median score is often zero or one out of 120.
A coder who did not know the hypotheses marked every moment a mathematician shifted attention from one inscription on the board to another, judged from gaze, gesture, speech, writing and erasing. That produced 4,653 attention shifts across 14 proof sessions. The core analysis covered six mathematicians, three men and three women, who each worked on one set theory problem and one geometry problem.
The researchers also flagged every time a mathematician said something like “Oh I see!” They found 27 such sudden insights in 13 of the 14 sessions.
Borrowing a tool from ecology
In physics and ecology, systems about to tip into a new state often show warning signs, such as growing fluctuations. The authors reasoned that a mind about to abandon a misconception might show something similar in its visible behavior.
Because blackboard work is a sequence of discrete moves rather than a single number, they measured surprisal: how unexpected each shift was given the mathematician’s recent pattern of moves. A simple model of insight they built showed that this measure should rise as understanding approaches a tipping point.
What happened before the aha
In the real recordings, behavior was significantly less predictable around insights. Average surprisal was 1.23 just before an insight and 1.25 just after, compared with 1.07 at other times. Over a session as a whole, predictability did not drift.
The rise began more than two minutes before the insight was spoken, climbed gradually, peaked just after the moment of insight and then fell sharply, as if the mathematician had settled into a new way of seeing the problem. A test that shuffled the timing of moves 500 times confirmed the effect was larger than chance.
One example makes it concrete. A mathematician who habitually moved her attention from a line representing an interval of numbers to a nearby list suddenly jumped instead to a triangle on the far side of the board. That rare move came about thirty seconds before her insight.
The authors argue that publication records are too coarse to explain individual breakthroughs, and that understanding insight means watching the hands, eyes and chalk of the person having it. Their method, they note, could apply to any system that produces a stream of discrete, symbolic events.
Study Details:
- Title: An information-theoretic foreshadowing of mathematicians’ sudden insights
- Authors: Shadab Tabatabaeian, Artemisia O’bi, David Landy, Tyler Marghetis
- Journal: Proceedings of the National Academy of Sciences, Vol. 122, No. 35
- Publication Date: August 18, 2025
- DOI: 10.1073/pnas.2502791122