By Matthew Warren

Philosophers and mathematicians have long held that maths can be aesthetically pleasing. “Mathematics, rightly viewed, possesses not only truth, but supreme beauty,” wrote Bertrand Russell, while Carl Friedrich Gauss proclaimed that “The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it”.

But a study published recently in Cognition suggests that even those whose lives don’t revolve around logic and numbers also have an appreciation for mathematical “beauty”. People tend to see similarities between mathematical proofs and certain paintings or pieces of music, the study finds, suggesting we all share an intuition for the aesthetics of mathematics.

In their first study, Samuel Johnson at University of Bath School of Management and Stefan Steinerberger at Yale University asked 300 online participants (200 of whom had not studied maths beyond school level) to read four mathematical arguments. These were written to be accessible even to lay people, and included proofs like those below. The one on the left illustrates that the sum of an infinite geometric series adds to 1, while the other on the right shows that adding consecutive odd numbers always gives a square number.

Participants had to rate on a scale from 0 to 10 how similar each of these arguments was to each of four landscape paintings, including as The Hay Wain by John Constable and A Storm in The Rocky Mountains by Albert Bierstadt (below).

The participants tended to agree on the similarity of the various arguments and paintings. For example, when the researchers ranked each participant’s 16 painting-argument ratings from most to least similar, there was significant overlap between participants. And the participants tended to see particular mathematical arguments as being most strongly related to a specific artwork (for example, overall they viewed the proof concerning the sum of consecutive odd numbers as most similar to the Constable painting).

This consensus among participants regarding how the maths arguments related to the artworks suggests that people share an intuition about the aesthetics of maths, the authors say – otherwise the similarity ratings would have been randomly distributed between participants. “Mathematical beauty, then, does not appear to be solely in the eye of the beholder but appears to have deep psychological roots,” they write.

The researchers repeated the experiment, replacing artworks with pieces of classical music, including the fugue from Bach’s Toccata in E Minor and Beethoven’s Diabelli Variations. Overall, participants’ ratings of the similarity of the maths and music were much higher than the maths/art similarity ratings in the previous study (the authors suggest that “music is imbued with a more mathematical character”), and again there was some consistency in ratings between participants.

Interestingly, similarity judgements were more consistent between groups with some level of mathematical training than between participants with and without much maths experience, suggesting that aesthetic judgments may change somewhat with expertise. (This idea was also supported by a third experiment, which suggested that participants with mathematical experience may use some additional criteria for judging the beauty of mathematical arguments compared to those without a maths background ).

The researchers have used a clever paradigm to demonstrate that even laypeople may assign some aesthetic quality to mathematical proofs – “almost as though they treat proofs as artworks”. But their experiments reveal frustratingly little about the broader relevance of assigning aesthetic values to abstract ideas. Some researchers have suggested that mathematicians use the “beauty” of an explanation as a guide to whether or not it is true, for example, and the authors seem to suggest that based on their results the same could hold for laypeople. However, they didn’t test this directly.

The way we use our aesthetic intuitions could have important implications. If people take into account the aesthetics of abstract ideas when making judgments about those ideas, then that suggests that perceived aesthetics could be a factor in everything from choosing between different scientific theories to making policy decisions, the researchers note. “We hope that this work serves as a broad call to action in investigating the aesthetics of abstract ideas,” they write.

—Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas

Matthew Warren (@MattbWarren) is Staff Writer at BPS Research Digest