Abstract mathematical ideas are as beautiful as works of art or music

A beautiful landscape painting, a beautiful piano sonata—those are words that describe aesthetics, beauty when it comes to art and music. What about math, is it possible to talk about an abstract idea with the word “beautiful”, in addition to the usual words like useful, intelligent?

Answer is possible. And that is not only the opinion of mathematicians but also ordinary people. That’s the conclusion of a study published this month in the international journal Cognition.

Abstract mathematical ideas are as beautiful as works of art or music
The beauty of mathematics is not just beauty in one dimension.

Accordingly, ordinary Americans (with no math training) rate mathematical formulas for their beauty in the same way that they judge works of art or music. The beauty of mathematics they recognized was not just beauty in one dimension. 300 participants were asked to rate specific beauty in 4 different mathematical formulas according to 9 aesthetic criteria such as elegance, complexity, universality….

The question of the aesthetics of mathematics began when Stefan Steinerberger, an assistant professor of mathematics at Yale University and co-author of the study, compared a proof he was teaching with “a really good Schubert sonata”. The math students at Yale also had an impressive amount of musical knowledge, so three or four people later went up to Stefan and asked him what the association meant. After an interview with the psychology department, Stefan was introduced to Samuel GB Johnson, study co-author and assistant professor of marketing at the University of Bath School of Management. Samuel is doing his doctoral thesis in psychology at Yale, studying reasoning and decision-making, how we humans evaluate different explanations and formulas about things.

The two then agreed on an experimental design to test Stefan’s hypothesis: do we share the same sensitivity to the beauty of mathematics as other fields like art or music, and in addition to Mathematicians, is this true for the average person?

The study used 4 mathematical formulas (proofs), 4 landscape paintings and 4 piano sonatas. The similarities between music and mathematics have long been noticed, so the authors wanted to test another aesthetic area, art, to see if there was something more universal in how we judge aesthetics.

Abstract mathematical ideas are as beautiful as works of art or music
Mathematical formulas are as beautiful as works of art or music.

The study consisted of three parts: In the first part, a group was asked to match four math formulas with four landscape pictures based on the aesthetic similarities they noticed. In part 2, another group matched math formulas with sonatas, and in part 3 another group independently voted each of the 4 artworks and 4 math formulas on 9 different criteria and total score for beauty on a scale of 0 to 10.

The evaluation criteria were taken from the mathematical beauty essay “A Mathematician’s Apology” by GH Hardy. GH Hardy, full name Godfrey Harold Hardy is a very famous British mathematician in the field of analysis and number theory with typical work being the Hardy-Littlewood conjecture. Hardy’s 6 criteria were further expanded to 9 criteria: seriousness, universality, depth, novelty, clarity, simplicity, elegance, complexity, and sophistication.

4 mathematical formulas used in the research are quite basic and easy to understand. Here is one of them.

Abstract mathematical ideas are as beautiful as works of art or music

A formula for the sum of a series of numbers geometrically used to evaluate mathematical beauty

The problem requires calculating the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ··· using geometry. We can prove this by cutting the square in the image above into pieces with a total area of 1.

The voting results of the group participating in part 3 show that artworks and math formulas with high scores on elegance will predict a high score on beauty. After looking at the similarity in the rating scores of groups 3 and 1 (which were asked to do the simpler task of matching mathematical formulas and artwork with similar aesthetics), the authors was surprised that group 3’s similarity was predictive of group 1’s results. Group 3 participants were more consistent when it came to scoring formulas and pictures of elegance, and group participants 1 has a similar tendency: the formulas and pictures considered by group 3 to be the most elegant are linked in the results of group 1.

Thus, ordinary people who do not study math not only have a similar perception of the beauty of mathematics and art, but also perceive the similarity between mathematics and art. In other words, they also agree on how to evaluate the elements that make up beauty, regardless of the field of beauty.

Abstract mathematical ideas are as beautiful as works of art or music
Artwork and math formulas that score high on elegance predict a high score on beauty.

Stefan hopes the research is done again with other musical, mathematical and artistic works. “We’ve demonstrated this phenomenon, but we don’t know its limits. It won’t exist anywhere? Does it have to be classical music? Does it need to be inherently natural pictures of the world? high aesthetics?”.

Both authors argue that this research has many implications in math education, especially at the secondary level. That would create “opportunities to make the increasingly abstract and formal aspects of mathematics more accessible and interesting to high school students” , and “could also be useful from an incentive perspective.” encourage more people to get into math” .

“Understanding what people think is beautiful in math will help you understand how people understood math in the first place and how they handled it” . And there are also potential human interpretations of this question, which is how we really think about things as humans. To understand this, we have an “obligation” to work with psychologists, adds Stefan.