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The Neuroscience of Mathematical Beauty

The views expressed are those of the author and are not necessarily those of Scientific American.


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Mathematics, rightly viewed, possesses not only truth, but supreme beauty” — Bertrand Russell

The latest neuroscience of aesthetics suggests that the experience of visual, musical, and moral beauty all recruit the same part of the “emotional brain”: field A1 of the medial orbitofrontal cortex (mOFC).

But what about mathematics? Plato believed that mathematical beauty was the highest form of beauty since it is derived from the intellect alone and is concerned with universal truths. Similarly, the art critic Clive Bell noted:

“Art transports us from the world of man’s activity to a world of aesthetic exaltation. For a moment we are shut off from human interests; our anticipations and memories are arrested; we are lifted above the stream of life. The pure mathematician rapt in his studies knows a state of mind which I take to be similar, if not identical. He feels an emotion for his speculations which arises from no perceived relation between them and the lives of men, but springs, inhuman or super-human, from the heart of an abstract science. I wonder, sometimes, whether the appreciators of art and of mathematical solutions are not even more closely allied.”

A new study suggests that Bell might be right. Semir Zeki and colleagues recruited 16 mathematicians at the postgraduate or postdoctoral level as well as 12 non-mathematicians. All participants viewed a series of mathematical equations in the fMRI scanner and were asked to rate the beauty of the equations as well as their understanding of each equation. After they were out of the scanner, they filled out a questionnaire in which they reported their level of understanding of each equation as well as their emotional experience viewing the equations.

This equation was most consistently rated as beautiful (Leonhard Euler’s identity):

Other winners of the equation beauty contest included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler’s formula for complex analysis, and the Cauchy-Riemann equations. In contrast, this equation was most consistently rated as ugly (Srinivasa Ramanujan’s infinite series for  1/π):

Other low-rated equations included Riemann’s functional equation, the smallest number expressible as the sum of two cubes in two different ways, and an example of an exact sequence where the image of one morphism equals the kernel of the next.

Looking at the brain scans, the researchers found that the experience of mathematical beauty was related to the same part of the brain that has been found in prior studies to be associated with the experience of visual, musical, and moral beauty: field A1 of the medial orbitofrontal cortex (mOFC). What’s more, the stronger the reported intensity of the experience, the stronger the brain activation in this area.

Interestingly,  understanding and beauty were far from perfectly correlated. Even the non-mathematical participants found some equations more beautiful than others, even though they didn’t understand the equations. The activation of field A1 of the mOFC was particularly related to ratings of beauty, not understanding.

Note that these findings do not mean that the mOFC was solely responsible for processing mathematical beauty. Other brain regions were also activated during processing the mathematical equations which are not recruited when viewing paintings or listening to musical excerpts. Instead, the findings suggest that the common factor when experiencing beauty across seemingly different stimuli is activation of field A1 of the mOFC.

Also, the results do not mean that this area of the brain is dedicated to processing beauty. Field A1 of the mOFC serves multiples functions, including processing emotion, learning, pleasure, and reward.

Finally, note that these results don’t mean that the cluster of neurons in field A1 of the mOFC caused the experience of mathematical beauty. The results merely indicate a correlation between the brain activation and the experience of beauty when viewing the math equations.

Nevertheless, the results are interesting and raise the intriguing question: If the experience of mathematical beauty is not necessarily related to understanding the equations, what is the source of mathematical beauty?

The researchers suggest that “there is an abstract quality to beauty that is independent of culture and learning.” They surmise that the non-mathematicians assessed beauty– not based on cognitive understanding– but based on formal qualities of the equations, such as the physical forms, symmetrical distribution, etc.

But the researchers go even further and suggest that beauty may be a pointer to what is true in nature. Indeed, Plato emphasized that mathematical formulations are experienced as beautiful because they give insights into the fundamental structure of the universe. Likewise, the theoretical physicist Paul Dirac noted that

“There is no logical reason why the (method of mathematical reasoning should make progress in the study of natural phenomena) but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme. . . What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. . . We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence.” (ellipses added)

Their findings also have implications for the philosophy of aesthetics, where it is often debated whether aesthetic experiences can be quantified, and the extent to which the experience of beauty is linked to pleasure and reward.

Lots of food for thought!

© 2014 Scott Barry Kaufman, All Rights Reserved.

h/t: Rebecca McMillan; image credit: united-academics.org

Scott Barry Kaufman About the Author: Scott Barry Kaufman is Scientific Director of The Imagination Institute in the Positive Psychology Center at the University of Pennsylvania. Follow on Twitter @sbkaufman.

The views expressed are those of the author and are not necessarily those of Scientific American.





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  1. 1. SJCrum 6:41 pm 05/7/2014

    As for this subject, the science of the soul brain is that it has three parts, with one being the emotional part, a second being that of intelligent thinking and analysis, and the third involving strong feelings for survival. These three, in the end, make up our “physical” soul brain. And, the soul brain is inside the brain inside our body.
    The point of this is that all of our emotional feelings are in the emotion brain part, and all of the intelligent thinking is in the other totally separate part.
    With that, it appears that the writer of this article is trying to describe the “ultra-enormous” emotional thrill that mathematicians and others get from looking at “something” that is totally on the wrong side of the brain.
    the point of this is that there is something that is farrrrrr more lusciously delicious than any of their scrumptiously yummy math things.
    GIRRRRRRRRLLLLS!!!!!!!!!!!!!!!!!!!!!!
    But, hey, each to his own thoughts about what is totally-out-of-this-world lusciously yummy, huh? You know, if you said “arf” to a dog, do you think it would look at you cluelessly as you pointed at the deliciously yummy female dog just to it’s furry right side?
    Yeah, well, a formula in a skimpy bikini ain’t worth beans in the real end of ultra-WOW.
    I am sorry, but I couldn’t resist this thought.

    Link to this
  2. 2. AUChrisG 7:07 pm 05/7/2014

    The formula for the normal distribution ain’t pretty. Take it from someone studying their first year of a Biostats Masters ;-)

    Link to this
  3. 3. SJCrum 4:33 pm 05/8/2014

    By the way, the 1 + e = 0 is actually a terrific equation that involves imaginary numbers, and that positive results can be accomplished by using it. Even its appearance is clean and somewhat “beautiful” in somewhat elegant way.
    HOWEVER, in real beauty there is a far greater and totally WOW equation that can totally knock the bloomers off of that one. So, you ask, “What fantastically great equation might that one be?”
    The ultra-fantastic equation is E = 1/2 MV^2, AND because of something truly magnificent that this equation describes. That, ultra-yummy gem is that it describes the physical process of making all atoms, and therefore, all of the totally luscious beauty of every physically delicious female body that will ever exist.
    Sooooooo, this equation at first appearance might not appear to be all that hot, but it is BY FAR the ULTRA-GREATEST, most enormously fantastic equation in the entire universe.
    And, (hee, hee), it beats the beans out of your relatively trivial in comparison, “e” thing.
    Sorry, but in truth, the other equation didn’t have one chance in zillions to be anywhere near as ultra, deliciously fantastic as mine here.
    Chicks, what can I say, they are totally lusciously yummy.

    Link to this
  4. 4. tanboontee 8:46 pm 05/9/2014

    Euler’s equation is indeed the most beautiful, mathematically speaking.

    Likewise, E = (mv^2)/2 could be among the most beautiful equations in Physics.

    Link to this
  5. 5. goatmaniac 4:39 pm 05/14/2014

    I like putting one and one together and getting eleven.

    Link to this

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