*[Mathematical beauty] “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,” Paul Dirac*

Mathematical formulae, like Euler’s identity, are rarely considered candidates for universal beauty like the works of Mozart, Shakespeare or Van Gogh. However, when mathematicians are shown equations while in a brain scanner, the same emotional brain centres used to appreciate art are activated.

A study by Proffesor Semir Zeki from University College London and co-workers^{(1)} published last week, shows that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain than the experience of beauty derived from other sources —field A1 of the medial orbito-frontal cortex. Their research suggests that there may be a neurobiological basis to beauty.

Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved. (Prof Semir Zeki, quoted by the BBC: “Mathematics: Why the brain sees maths as beauty”)

This is very interesting because, since a long time ago, the experience of beauty derived from mathematics has been thought to represent the most extreme case of the experience that is dependent on learning and culture. Plato for example thought that:

Nothing without understanding would ever be more beauteous than with understanding, and further that understanding cannot arise anywhere without a soul.

Hence, the authors have paid special attention to this relationship and they conclude that the correlation between beauty and understanding, though significant, is imperfect. Some combinations of form are more aesthetically pleasing than others, even if they are not “understood” cognitively. There seem to be an abstract quality to beauty that is independent of culture and learning. Euler beats Ramanujan^{(2)}.

Plato also considered that the experience of mathematical beauty is the highest form of beauty, since it is derived from the intellect alone and it is concerned with eternal and immutable truths. He pointed to the capital question of whether beauty, even in such an abstract area as mathematics, is a clue to what is true in nature. This has been a question of intense debate, especially among physicists, as they pursued ever more sophisticated theories which, in many cases, possess an “unreasonable” explanatory power.

The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. (Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”)

General theory of relativity is a case in point. It is widely recognized as one of the most beautiful physical theories, and Einstein seems to have been guided in its formulation by a fundamental if “unnecessary” aesthetic instinct or emotion:

… it would have been entirely sufficient if Einstein had sought a theory that would allow for such small deviations from the predictions of the Newtonian theory by a perturbative treatment. That would have been the normal way. But that was not Einstein’s way: he sought, instead, an exact theory. And his only guides in his search for an exact theory were the geometrical base of his special theory of relativity provided by Minkowski and the principle of equivalence embodying the equality of the inertial and the gravitational mass. (Subrahmanyan Chandrasekhar, “Beauty and the Quest for Beauty in Science”)

Paul Dirac also acknowledged that what made the theory of relativity so acceptable to physicists, in spite of its going against the principle of simplicity, was its great mathematical beauty. He wondered why mathematical reasoning is so effective in the study of Nature:

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… (Paul Dirac)

Dirac himself was guided by the aesthetic instinct, as he explained in this interview conducted by David Peat in the early 1970’s:

David Peat:The papers you produced have been universally considered beautiful. Were you guided by notions of beauty?

Paul Dirac:Very much so. One can’t just make random guesses. It’s a question of finding things that fit together very well. You’re solving a problem, it might be a crossword puzzle, and things don’t fit, and you conclude you’ve made some mistakes. Suddenly you think of corrections and everything fits. You feel great satisfaction. The beauty of the equations provided by nature is much stronger than that. It gives one a strong emotional reaction.

Murray Gell-man have also tacked this strange connection between beauty and truth in fundamental physics:

…we have this remarkable experience in this field of fundamental physics that beauty is a very successful criterion for choosing the right theory. And why on earth could that be so? (Murray Gell-man, “Beauty, truth and … physics?”)

The truth is that we have not the slightest idea. However, truth is so elusive that I can recommend you that, when in doubt, bet on unreasonable beauty. So did Herman Weyl:

My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful,” (Hermann Weyl’s Legacy)

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(1) Zeki S, Romaya JP, Benincasa DMT and Atiyah MF (2014) “The experience of mathematical beauty and its neural correlates” Front. Hum. Neurosci. 8:68. doi:10.3389/fnhum.2014.00068

(2) The formula most consistently rated as beautiful was Leonhard Euler’s identity. It links 5 fundamental mathematical constants with three basic arithmetic operations each occurring once. The one most consistently rated as ugly was Srinivasa Ramanujan’s infinite series for 1/π.