"Do you think math has an aesthetic appeal?" If you were to interview this question randomly on the street, most people would probably say "don't understand." If you go to middle or high school and ask those middle school students, you're likely to get more negative answers.
But a study by semil ZEK, a neuroscientist at the University of London, UK, gives a surprising answer: the beauty of mathematics, at least for mathematicians, is consistent with the beauty of literature and art. When a mathematician sees an aesthetic mathematical formula, the level of activation in the ventral orbitofrontal cortex (located in the anterior part of the human brain) is equivalent to that of a person who appreciates a beautiful work of art or beautiful music. The orbitoprefrontal cortex is a very important part of the pleasure experience and reward circuit. The study also found that the stronger the aesthetic of the formula, the higher the level of activity in the brain area. So, maybe our brains can give an answer to whether this mathematical formula is beautiful or not.
Beauty or not, the brain knows
As early as 1999, Semirzek proposed the concept of neuroaesthetics in his book Inner Vision: Exploring the Relationship between Art and the Brain. His dedication to exploring the psychological and neural activity of humans in appreciating and creating works of art has sparked a wave of research on the relationship between aesthetic processes and neural activity in the brain. Zeke also conducted a series of experiments to examine brain activity during aesthetic experience.
In 2016, Zeke and Shi published a study in the Journal of Public Science Libraries Synthesis, focusing on people's aesthetic feelings when viewing works of art. The research team first recruited 30 volunteers and asked them to rate 60 pieces of music and 60 paintings on a scale of 1-9. Among them, 1 is "ugly", 9 is "beautiful", and 5 is "insensitive". Based on this rating, 30 musical compositions and 30 paintings were selected. One-third of them are non-aesthetic and ugly works, one-third are beautiful works, and one-third are neutral works. On this basis, another group of volunteers was invited to do an MRI experiment. They had volunteers lie down on an MRI machine and listen to selected music or paintings. After listening to a piece of music or seeing a piece of work, they were asked to rate the stimulus, whether it was "beautiful", "not feeling", or "ugly". The results showed that when volunteers heard beautiful music or saw beautiful artwork, their ventral orbitofrontal cortex and caudate nucleus activity levels were higher, and the activity level of orbitofrontal cortex was consistent with the volunteers' ratings of these works, that is, the higher the subjective score, the stronger the brain activity. The orbitofrontal cortex is an important brain region in the reward circuit. When we are rewarded, such as eating delicious chocolate or receiving a monetary reward, the activity level of this brain region increases.
As far as works of art are concerned, taking painting as an example, the purpose of some paintings is to "seek truth", that is, to reflect the attributes of the objective world; on the other hand, the process of artistic creation is also the process of concretizing the artist's own ideals and emotions, so this is also the process of emotional expression. As far as aesthetic experience is concerned, it also consists of two parts: one is the intrinsic characteristics that come from the stimulus (objective beauty), and the other is from the individual emotional experience (subjective beauty). Therefore, the aesthetic experience is to a certain extent independent of the work of art and reflects the subjective feelings of the individual. Especially for non-art students, aesthetic judgment comes from the emotional experience of the individual, which is closely related to the reward system.
The beauty of mathematics in the minds of mathematicians
The aesthetic experience of seeing mathematical formulas mentioned at the beginning of this article was a research report published by ZEK's research team in 2014 at the frontier of human neuroscience. They recruited 16 participants (graduate or postdoc) with expertise in mathematics and included 60 formulas in the study, 20 of which were in the "ugly" category, 20 in the "beautiful" category, and 20 in the "neutral" category.
The results show that when a person sees an aesthetic formula (such as Euler's formula, eπI+1=0), mathematicians call it "the formula created by God." It connects the most important numbers in mathematics, including the basis e, PI, the imaginary unit I, the 1 unit of the natural number, and 0, which is commonly used in mathematics), and the ventral orbitofrontal cortex is highly activated. In addition, an individual's brain activity has no relationship with the individual's understanding of the formula, because the individual's understanding of the formula mainly affects the activity level of the visual cortex, which is higher when the individual sees an unfamiliar formula.
The research team sought to recruit a group of non-math experts as a control group. The results of the preliminary behavioral study showed that out of the 720 mathematical formulas provided in the study, the individual's comprehension of 720 mathematical formulas was 0, which met the inclusion criteria. However, for another open question: "When you see a beautiful formula, do you feel that there will be an emotional experience associated with it?" three-quarters answered in the negative. Although sometimes they think that a formula is aesthetic, it is also based on the form of the formula (like symmetry, length, etc.), that is, the formula just looks appealing. So in the end, the researchers didn't do an MRI scan to measure their brain activity.
But for mathematicians, the beauty they experience is more visceral. To the question of emotional experience mentioned above, all mathematicians answered in the affirmative, mentioning "a little excited", "like hearing good music or seeing an oil painting that attracted me", "I feel like my heart will jump out". Also, to the next question: "Do you experience happiness, happiness and fulfillment when you see the beautiful formula?" all the participants also answered in the affirmative. Therefore, for these math professionals, the well-being they experience goes beyond the beauty of the mathematical formula itself and merges with the emotional feelings of the individual, which further validates the important role of the reward ring in the aesthetic experience.
Art and mathematics sometimes seem like two extremes, the former seems to be more emotional and more accessible to the average person, and as for mathematics, it has a sense of distance and mystery due to its depth and complexity that requires deep study to truly understand it. However, the appreciation of the beauty of mathematics also runs through the ages. Plato, Russell, and Dirac were all fascinated by the beauty of mathematics.
Maslow, an American social psychologist, first proposed the hierarchy of needs theory, which only includes five levels: physiological needs, security needs, belonging and love needs, self-esteem needs and self-actualization needs. These five levels of need are the source of human motivation. Later, Maslow expanded this model to include the need for knowledge and beauty before the need for self-actualization. However, he pointed out that the satisfaction of the demand for beauty, as one of the high-level needs of human beings, is a growth need, and people's demand for this need is not as strong as the basic needs. At that time, people did not know that the satisfaction of the need for beauty would bring us the same pleasure experience as when the physical need was satisfied, and that there was a neural basis for this pleasure experience. The satisfaction of the need for beauty may be the reason why we love and pursue art, and it is also the reason why some artists or mathematicians devote themselves to creation or research.