For some people, math can be a necessary headache. Yes, algorithms guide countless aspects of daily life. There are tips to calculate and hours to count. But unless someone's a specialist, they'll probably ignore complex math in any given situation if they can help it.
But Yale assistant professor of mathematics Stefan Steinerberger wants to challenge that perception. His new study shows that an average American can assess mathematical arguments for beauty just as they can pieces of art or music.
And he has the numbers to prove it.
Beauty is in the eye of the beholder, of course. But for Steinerberger and coauthor Samuel G.B Johnson, beauty is made up of nine separate components: seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy, and sophistication. They didn't come up with those criteria themselves, but expanded on ideas laid out in “A Mathematician’s Apology,” a 1940 essay by mathematician G.H. Hardy.
"The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics," Hardy wrote in his essay, which meant to draw distinctions between applied mathematics, as seen today in computer science and statistics, and what he called "pure," or theoretical, mathematics.
Steinerberger's interest in the concept of mathematical beauty came when he compared a proof he was teaching to a “really good Schubert sonata.”
“As it turns out, the Yale students who do math also do a statistically impressive amount of music,” Steinerberger said in a press release. “Three or four students came up to me afterwards and asked, ‘What did you mean by this?’ And I realized I had no idea what I meant, but it just sounded sort of right. So, I emailed the psych department.”
The hunt of understanding beauty eventually led him to Johnson, now an assistant professor of marketing at the University of Bath School of Management, whose specialities are quite different than Steinerberger's. “A lot of my work is about how people evaluate different explanations and arguments for things,” he said.
The two realized they were a good match, and decided to analyze aesthetic sensibilities within math as happens with other forms like art or music.
“I had some diffuse notion about this, but Sam immediately got it,” said Steinerberger. “It was a match made in heaven.”
One of the mathematical arguments used in the study.
Their study was broken into three segments. As described by Yale, first it required a "sample of individuals to match the four math proofs to the four landscape paintings based on how aesthetically similar they found them; the second required a different sample to do the same but instead comparing the proofs to sonatas; and the third required another unique sample of people to independently rate, on a scale of zero to ten, each of the four artworks and mathematical arguments along nine different criteria plus an overall score for beauty."
If this all sounds pretty elaborate, remember: It's a study of beauty by a mathematician.
But the results speak for themselves. People with no professional involvement in math or the arts held "similar intuitions about the beauty of math as they did about the beauty of art," they say.
“I’d like to see our study done again but with different pieces of music, different proofs, different artwork,” said Steinerberger. “We demonstrated this phenomenon, but we don’t know the limits of it. Where does it stop existing? Does it have to be classical music? Do the paintings have to be of the natural world, which is highly aesthetic?”
The pair see educational prospects in their work. If proofs can be seen as beautiful, then students more curious about art than numbers might be able to see the similarities if presented with them.
“There might be opportunities to make the more abstract, more formal aspects of mathematics more accessible and more exciting to students at that age,” said Johnson, “And that might be useful in terms of encouraging more people to enter the field of mathematics.”
