The beauty of mathematical formulations resides in abstracting, in simple equations, universal truths. Many people, including mathematicians Bertrand Russell (1919) and Hermann Weyl (Dyson, 1956; Atiyah, 2002), physicist Paul Dirac (1939), and art critic Clive Bell (1914), have written about the importance of beauty in mathematical formulations and compared the experience of mathematical beauty to that of the greatest works of art (Atiyah, 1973).
Their descriptions imply that the experience of mathematical beauty is similar to that of other sources, despite the fact that mathematical beauty stems from a far deeper intellectual source than visual or musical beauty, which are more “sensible” and perceptually grounded. Previous research into the neuroscience of beauty has revealed that the sense of visual, musical, and moral beauty all connect with activity in a specific region of the emotional brain.
The researchers utilized functional magnetic resonance imaging (fMRI) to record the brain activity of 15 mathematicians as they saw mathematical formulas that they had previously classified as beautiful, neutral, or ugly. The findings revealed that the sense of mathematical beauty corresponds with activity in the same area of the emotional brain as the experience of beauty derived from art or music, specifically the medial orbito-frontal cortex.
“To many of us mathematical formulae appear dry and inaccessible but to a mathematician an equation can embody the quintescence of beauty. The beauty of a formula may result from simplicity, symmetry, elegance or the expression of an immutable truth. For Plato, the abstract quality of mathematics expressed the ultimate pinnacle of beauty.”Professor Semir Zeki
This raises the question of whether the sense of beauty obtained from a highly intellectual and abstract source, such as mathematics, corresponds with activity in the same area of the emotional brain as that generated from more sensory, perceptually based sources.
Each subject in the research was given 60 mathematical formulas to evaluate at their leisure and score on a scale of -5 (ugly) to +5 (beautiful) based on how beautiful they perceived them to be. They were asked to re-rate them two weeks later while in an fMRI machine.
Leonhard Euler’s identity, the Pythagorean identity, and the Cauchy-Riemann equations were consistently assessed as beautiful (both before and after the scanning). Leonhard Euler’s identity connects five fundamental mathematical constants with three basic arithmetic operations, each occurring once, and its beauty has been compared to that of Hamlet’s soliloquy. The ugliest were Srinivasa Ramanujan’s infinite series and Riemann’s functional equation, according to mathematicians.
“We have found that, as with the experience of visual or musical beauty, the activity in the brain is strongly related to how intense people declare their experience of beauty to be – even in this example where the source of beauty is extremely abstract. This answers a critical question in the study of aesthetics, one which has been debated since classical times, namely whether aesthetic experiences can be quantified.”
The experience of mathematical beauty and its neural correlates, Semir Zeki, John Paul Romaya, Dionigi M. T. Benincasa, and Michael F. Atiyah
Published: February 2014