Scientists and mathematicians often describe facts, theories and proofs as “beautiful,” even using aesthetics to help guide their work. Their criteria might seem opaque to nonexperts, but new research finds that novices can consistently assess a proof's beauty or ugliness.
A mathematician and a psychologist analyzed responses from about 200 online participants for each of three experiments in their study, published in August in Cognition. Most had attended college but had not studied math beyond university calculus. In each experiment, they read four simple mathematical arguments and were tested for comprehension. (Two included diagrams; see the graphic for an example.)
The subjects rated each argument's “similarity” to each of four landscape paintings, and the results were clearly consistent: people generally agreed on which arguments matched which paintings—and their choices roughly aligned with those made by eight mathematicians. (The argument pictured was most strongly matched to a Yosemite landscape by Albert Bierstadt, seen here.) The second experiment produced a similar result with classical piano music in place of paintings.
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Credit: Jen Christiansen; Source: “Intuitions about Mathematical Beauty: A Case Study in the Aesthetic Experience of Ideas,” by Samuel G. B. Johnson and Stefan Steinerberger, in Cognition, Vol. 189; August 2019
For the third experiment, subjects rated the arguments and paintings on 10 adjectives, including “beautiful.” Again, results were consistent, and elegance, followed by profundity and clarity, was the biggest factor in judging beauty for both the math and the art. Samuel Johnson, a psychologist at the University of Bath in England, and one of the paper's co-authors, says he was most surprised that those qualities could predict the first group's pairings of ideas and paintings—indicating that the math-art correspondence was based on something deeper than superficial geometric features.
“It's a very clever study,” says Matthew Inglis, a researcher in math education at Loughborough University in England, who was not involved with the work. “I found the results to be quite counterintuitive, albeit very persuasive. Based on my own work”—in which mathematicians disagreed about the quality of proofs—“I would have expected aesthetic judgments in mathematics to be unstable across individuals.”
Nathalie Sinclair, a math education researcher at Simon Fraser University in British Columbia, who was also not involved in the study, was surprised as well. “One might have thought that because there is so much fear of mathematics in our culture, people would have thought the questions were absurd,” she says.
Stefan Steinerberger, a mathematician at Yale University and a paper co-author, believes educators should highlight the beauty in math. “People have this weird notion of thinking of themselves as incorporeal computing machines,” he says. “It's not true.”