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Researchers have developed a model that predicts when mathematicians are on the cusp of a breakthrough, something not even the individuals themselves are capable of.
Videos of expert mathematicians working to solve difficult maths problems revealed their behaviour become significantly more unpredictable in the minutes leading up to a “eureka” moment.
“Mathematical insights involve sudden and unexpected transitions from confusion to clarity. Mathematicians describe these ‘aha!’ or ’eureka’ insights as genuinely surprising, and afterward they are often unable to explain how they arrived at their insight,” write the authors of a study presenting the findings in Proceedings of the National Academy of Sciences.
“The suddenness of ‘aha’ insights in mathematics is reminiscent of critical transitions – abrupt shifts from one stable regime to another – that occur in a variety of other complex systems, from motor control and mental health to natural ecosystems and global climate.”
These critical transitions, they say, are sometimes anticipated by signals that indicate the system is unstable and at risk of tipping into a new regime.
“We borrowed theory and methods from statistical physics and theoretical ecology to attempt to foreshadow the onset of these sudden insights,” the authors write.
To do this, the researchers recorded videos of 6 expert mathematicians solving problems from the William Lowell Putnam Mathematical Competition.
The annual competition for undergraduates in the US and Canada “lasts 6 hours, consists of 12 problems, and in most years the median score is 0 or 1 out of 120,” the authors note.
In each session, they noted how the mathematicians interacted with their blackboards, including writing and gesturing. Though the experts began each proof session confused, they arrived at at least 1 epiphany.
“We identified every moment when mathematicians expressed in speech a sudden insight (e.g. ‘Oh I see!’). These moments were often accompanied by sudden movements and emotional exclamations, not unlike Archimedes running naked through the streets of ancient Syracuse yelling ‘eureka’, though admittedly less dramatic,” the authors write.
“Presumably mathematicians also had other, perhaps smaller, insights that were not expressed verbally.”
The researchers developed a model which identified that insights which “appear to come out of nowhere are actually prefigured by changes in how mathematicians are writing and gesturing”.
It found that their behavioural “unpredictability” increased in the 2 minutes leading up to the eureka moments and peaked at about 1 minute following an insight, before returning to baseline.
“In the minutes leading up to an insight, mathematicians’ shifts of attention between inscriptions enacted connections that were increasingly unprecedented,” the authors write.
“This kind of associative or combinatorial thinking is a hallmark of creativity, including in scientific arenas such as biomedical research and technological innovation … We captured this combinatorial process as it unfolded moment-to-moment in mathematicians’ embodied activity.”
The researchers say that a variety of factors might explain this increase in unpredictable activity.
For example, a period of little progress might exacerbate a mathematician’s frustration, increasing their openness to innovation or allowing other ideas to come forward.
It may also reflect a conscious choice to intentionally explore new connections or “experiment and break with traditional pathways of investigation”.
“At a biological level, this increase in behavioural unpredictability might reflect a destabilisation of underlying brain dynamics,” they speculate.
“Future work could investigate whether sudden mathematical insights are foreshadowed by the destabilisation of brain areas thought to be specialised for mathematical thought.”