Thursday, April 30, 2015

The Experience of Mathematical and Biological Beauty

Contributed by Mikhail Filippov and Semir Zeki

The experience of mathematical beauty is perhaps the extreme case of the experience of beauty conditioned by culture and learning. One cannot experience mathematical beauty unless one is mathematically cultured. Those not versed in mathematics are unlikely to experience beauty in equations that mathematicians find beautiful and sometimes are even moved by.

And yet, mathematical language is universal. Mathematicians of different culture – from China, Western Europe, Africa, Russia – are able to experience beauty in the same equations even in spite of their profound cultural and linguistic differences. Hence, in another sense, mathematical beauty is not conditioned by culture and language. Indeed, it can perhaps be said that mathematical beauty is less culturally biased than other forms of beauty.

We must seek its source elsewhere than in cultural differences.

An interesting article on the experience of mathematical beauty suggests that Immanuel Kant saw the source of the beauty in mathematical equations in the fact that “they make sense”.

This raises two questions:  what does it “make sense” to, and why does it make sense to people of different cultures, who are nevertheless apart from people of all cultures, even their own,  who are not versed in the language of mathematics.

Our answer is that it makes sense to the logic of the brain, in that it is consistent with the logic that has evolved in the brain. The implication is obviously that the logical system of the brain is similar to those from different cultures. In other words, these mathematical equations make sense to people of different cultures because the logic of the brain is similar, in spite of cultural differences. Hence the common experience of beauty in the same equations reveals something about that logic.

It is therefore not surprising to find that there was, in our sample of mathematicians, a fair consensus in rating Leonard Euler’s identity formula,

                                    1 + ei = 0  

which links 5 fundamental mathematical constants with three basic arithmetic operations, each occurring once, as very beautiful.

But is this significant uniformity in rating an equation as beautiful vastly different from the rating of visual beauty by subjects belonging to different cultures? It is a subject worth addressing. 

We surmise that, if subjects from different cultures were asked to rate what we broadly call “biological” stimuli, such as human faces and bodies, in terms of beauty, there would also be a fair consistency. We also surmise that there will be a similar consistency when subjects from different cultures rate human faces and bodies as ugly. This consistency may not be apparent when subjects are asked to rate the beauty of artefactual stimuli, such as buildings; here culture and learning may play a more significant role.

Hence the experience of what we broadly refer to here as “biological” beauty, beauty in art, may be dictated by inherited brain concepts of what is “right” and makes sense, just as in mathematical formulae what is experienced as beautiful makes sense. Both, in other words, fall into a biological category, which distinguishes them from the beauty dictated by acquired, synthetic, brain concepts, as in the experience of architecture as beautiful. Acquired brain concepts are more conditioned by culture and learning and are hence are modifiable throughout life. Mathematical beauty is more resistant to cultural influences.

The experience of mathematical and biological beauty, even in spite of the fact that the former depends upon learning and the latter does not, therefore share, paradoxically, a similarity in that both are dictated by inherited brain concepts which makes them impervious to cultural differences but which, in the case of mathematics, can only be revealed through a language – that of mathematics – that individuals must acquire before the experience can be enabled.

This of course raises the question of what the logic of the brain represents and how it developed and evolved? Was it in response to the structure of the Universe, as Plato in ancient Greece and Paul Dirac in more modern times, would claim?

These are problems worth thinking about.