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We like our math like we like our art: beautiful

We like our math like we like our art: beautiful

Is it possible for a concept to be beautiful? Mathematicians frequently describe arguments as “beautiful” or “dull,” and prominent scientists have argued that mathematical beauty serves as a pointer to the truth. Do laypeople, like mathematicians and scientists, have an aesthetic experience with mathematics? According to three studies, they do. When participants assessed the resemblance of basic mathematical arguments to landscape paintings or pieces of classical piano music, their rankings were internally consistent.

The study, co-authored by a Yale mathematician and a psychologist from the University of Bath, demonstrates that typical Americans can evaluate mathematical arguments for beauty just as they do pieces of art or music. The math’s beauty, they discovered, was not one-dimensional either: using nine criteria for beauty — such as grace, complexity, universality, and so on — 300 people agreed on the particular ways that four distinct proofs were beautiful.

One of the four mathematical arguments used in the study, as it was displayed to participants.
One of the four mathematical arguments used in the study, as it was displayed to participants.

Johnson split the research into three sections. The first task required a sample of people to match the four math proofs to the four landscape paintings based on how aesthetically similar they found them; the second task required a different sample to do the same but compare the proofs to sonatas, and the third task required another unique sample of people to independently rate each of the four artworks and math proofs on a scale of zero to ten.

They developed these criteria from G.H. Hardy’s 1940 article “A Mathematician’s Apology,” which examines mathematical beauty. The nine qualities developed by the researchers from Hardy’s six were: seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy, and sophistication. The next stage was to compute the “similarity scores” for the group three participants, which indicated how aesthetically similar they thought each argument and artwork were to each other based on the different beauty criteria.

When the data were in, they were able to use the similarity scores from the third test to predict how participants would react in the first task. Participants in the third group agreed on which arguments were elegant and which paintings were beautiful, but participants in the first group preferred to connect the most elegant argument with the most elegant artwork.

Not only did laypeople have comparable intuitions about the beauty of arithmetic as they did about the beauty of art, but they also had similar intuitions about beauty. In other words, there was agreement on what constitutes beauty, regardless of medium.

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“I think if you understand what people consider beautiful in math, then it could give insight into how people understand math in the first place and how they process it,” added Steinerberger. “There’s also the human implication of the question: How are we actually thinking about things as human beings? I think we have an obligation to collaborate with psychologists on this.”

Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas, Samuel G.B. Johnson, StefanSteinerberger

Published: August, 2019
https://doi.org/10.1016/j.cognition.2019.04.008

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