## Maths makes your brain light up… but why?

By now you may have spotted the news story that the BBC headlined as “Mathematics: Why the brain sees maths as beauty”. As usual with science reporting, the headline doesn’t quite capture what the story’s about. The story’s based on a recently published paper by Zeki, Romaya, Benincase and Atiyah titled “The experience of mathematical beauty and its neural correlates”, and the key result of the study it describes is that when mathematicians look at an equation they regard as “beautiful”, there is activity in the same part of the brain that “lights up” when we look at other beautiful objects. Informally, we can conclude that mathematical beauty is experienced in a similar way to other forms of beauty — and perhaps that maths isn’t such an abnormal pleasure after all…

The famous dead salmon experiment should make us sceptical about reading too much into neuroimaging, but in justice to these researchers they seem to have been very careful in their analysis and they’re also careful not to overstate what they claim. Two aspects of the work, apart from the headline result, might be of particular interest to mathematicians. One is the question of which equations were regarded as particularly beautiful; the other is the relationship between understanding an equation and appreciating its beauty.

The complete list of equations used in the experiments is available as “Data Sheet 1” from the sidebar of the online article, and their scores in the pre-experiment beauty contest are in the spreadsheet “Data Sheet 3”. (You mght like to take a look at these and see how you rate them…) The top-rated equation was an old classic, Euler’s identity

$1 + e^{i\pi} = 0$.

Also jostling for positions on the podium were the Pythagorean identity

$\cos^2\theta+\sin^2\theta = 1$,

Euler’s expression for the complex exponential

$e^{i\theta} = \cos\theta + i\sin\theta$

(of which Euler’s identity is just a special case!), and the Cauchy–Riemann equations

$\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}\quad \mathrm{and}\quad\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$.

Firmly in the relegation zone, on the other hand, were Euler’s polyhedron formula (sorry Leonhard: can’t win them all)

$V-E+F=2$,

$\displaystyle\int_MK\,dA + \displaystyle\int_{\partial M}k_gds = 2\pi\chi(M)$,

and the spectral theorem

$A = \displaystyle\int_{\sigma(A)}\lambda\,dE_{\lambda}$,

while right at the bottom of the table was one of Ramanujan’s bizarre formulae for $\pi$,

$\dfrac{1}{\pi} = \dfrac{2\sqrt{2}}{9801}\displaystyle\sum_{k=0}^{\infty}\dfrac{(4k)!(1103+26390k)}{(k!)^4396^{4k}}$.

I found myself less surprised by the top of the league table than by the bottom of it: Euler’s identity regularly tops lists of the most beautiful equations out there, while it as well as Pythagoras’s theorem and the complex exponential connect elementary mathematical objects in a very fundamental (and useful) way. The beauty of the Cauchy–Riemann equations is perhaps less apparent until you’ve done a first course in complex analysis and come to realise just how strongly they structure the behaviour of analytic functions.

At the bottom, I guess Euler’s polyhedron formula just looks too simple to tickle many people’s fancy — which is a shame for the philosophers of mathematics, because the derivation of this formula is a classic (and much-debated) case study in how mathematical results are obtained. I was surprised to see Ramanujan’s formula down there, because for me such expressions pack a considerable punch due to their sheer unexpectedness… but it’s possible that Ramanujan’s “strange Indian melodies”, as Douglas Hofstadter called them, strike the brain rather more like one of Courtney Pine’s further-out solos than one of Mozart’s sonatas — that is to say, our initial reaction is puzzlement, followed by appreciation, rather than pure enjoyment from the outset.

The Gauss–Bonnet and spectral theorems, I suspect, suffered from a different handicap, which is that they are sufficiently “technical” that many mathematicians (myself included) simply don’t know their context well enough to appreciate them. This brings us to the second interesting aspect of the study: it seems that to appreciate mathematical beauty it really helps if we understand what we’re looking at.

When non-mathematicians were shown the equations, they identified some of them (though not many!) as more beautiful than others, but their judgement didn’t coincide closely with that of the mathematicians — the researchers conclude that they “did so on the basis of the formal qualities of the equations”, or to put it more colloquially, whether they made pretty shapes on the page. For mathematicians, on the other hand, there was a significant correlation between their understanding of an equation and how beautiful they rated it as being. The intellectual beauty that we experience in mathematics really does seem to be earned by our knowledge of mathematics — to borrow a line from Plato, quoted by the researchers, “nothing without understanding would ever be more beauteous than with understanding”.

Nevertheless, the correlation between beauty and understanding wasn’t perfect. Our experience of mathematical beauty, it seems, is more than just a sense of satisfaction that we’ve understood something: understanding, perhaps, acts only as a gateway to appreciation, and exactly what else contributes to that appreciation lies beyond the ken of neuroimaging. It seems fitting that the paper ends, like all honest discussions of aesthetics, on an open note:

… that there was an imperfect correlation between understanding and the experience of beauty and that activity in the mOFC [the relevant bit of the brain] cannot be accounted for by understanding but by the experience of beauty alone, raises issues of profound interest for the future. It leads to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved…

Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.

And possibly that is all we can know for the moment, even if it’s not quite all we need to know

(DP)