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
= 0
which links 5 fundamental mathematical constants with three
basic arithmetic operations, each occurring once, as very beautiful.
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.
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