According to a series of experiments by Yale University and University of Bath researchers, ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a piano sonata.
Mathematicians often describe arguments as beautiful or dull, and famous scientists have claimed that mathematical beauty is a guide toward the truth. Do laypeople, like mathematicians and scientists, experience mathematics esthetically? Three studies by Johnson & Steinerberger suggest that they do. Image credit: Gerd Altmann.
The similarities between mathematics and music have long been noted but Yale mathematician Stefan Steinerberger and University of Bath psychologist Samuel Johnson wanted to add art to the mix to see if there was something universal at play in the people judge esthetics and beauty.
The researchers designed experiments to test their question of whether people share the same esthetic sensibilities about maths that they do about art or music — and if this would hold true for an average person, not just a career mathematician.
They chose four mathematical proof, four landscape paintings, and four classical piano pieces. None of the participants was a mathematician.
The mathematical proofs used were: the sum of an infinite geometric series, Gauss’ summation trick for positive integers, the Pigeonhole principle, and a geometric proof of a Faulhaber formula.
The piano pieces were Schubert’s Moment Musical No. 4, Bach’s Fugue from Toccata in E Minor, Beethoven’s Diabelli Variations and Shostakovich’s Prelude in D-flat major.
The landscape paintings were Looking Down Yosemite Valley, California by Albert Bierstadt; A Storm in the Rocky Mountains, Mt. Rosalie by Albert Bierstadt; The Hay Wain by John Constable; and The Heart of the Andes by Frederic Edwin Church.
The first task required a sample of individuals to match the four maths proofs to the four landscape paintings based on how esthetically similar they found them.
The second task required a different group of people to compare the four maths proofs to the four piano sonatas.
Finally, the third asked another sample group to rate each of the four works of art and mathematical arguments for nine different criteria: seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy, and sophistication.
Participants in the third group agreed with each other about how elegant, profound, clear, etc., each of the mathematical arguments and paintings was.
But the scientists were most impressed that these ratings could be used to predict how similar participants in the first group believed that each argument and painting were to each other.
This finding suggests that perceived correspondences between maths and art really have to do with their underlying beauty.
Overall, the results showed there was considerable consensus in comparing mathematical arguments to artworks. And there was some consensus in judging the similarity of classical piano music and mathematics.
“Laypeople not only had similar intuitions about the beauty of math as they did about the beauty of art but also had similar intuitions about beauty as each other,” Dr. Johnson said.
“In other words, there was consensus about what makes something beautiful, regardless of modality.”
However, it was not clear whether the results would be the same with different music.
“I’d like to see our study done again but with different pieces of music, different proofs, different artwork,” Dr. Steinerberger said.
The findings were published in the journal Cognition.
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Samuel G.B. Johnson & Stefan Steinerberger. 2019. Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas. Cognition 189: 242-259; doi: 10.1016/j.cognition.2019.04.008
