MATHEMATICAL BEAUTY FACING PHILOSOPHICAL SCEPTICISM
University of São Paulo
Philosophers of mathematical practice have recently returned to the intriguing question of
mathematical beauty. This clearly does not belong to traditional aesthetics, because of the
abstractness of mathematical matter. Interestingly, mathematicians insist on its presence in
mathematics and seem to grant it an important status. They regard mathematical beauty as a
major inspiration and often a factor motivating their choices and preferences in practice, such
as the search for more elegant proofs and solutions. Philosophers try to find an account for this
very special attitude. Sceptics trumpet the fact that mathematics has nothing to please our
senses, and claim that all intellectual pleasures are simply epistemic, hence non-aesthetic.
This talk aims to introduce the audience to the recent discussion, present the main actors and
the main lines of play. Then it suggests a new angle on the situation from which something new
can be learned to defeat a sceptic.
The first suggestion is to widen the focus from proofs, to include mathematical structures.
Proofs have a major justificatory purpose, whereas structures do not have any epistemic
purpose; so it is credible that they can have properties which mathematicians value
independently of epistemic function. This opens the possibility of mathematical beauty at least
for mathematical structures. Then, if we can make a good case for claiming that it is possible
for an abstract mathematical structure to be beautiful, we can return to proofs to investigate
whether they too can have positive aesthetic value, without having to worry that aesthetic
value is restricted to concrete sensory entities. The second suggestion is about the choice of
mathematical entities to look for instances of mathematical beauty. It is instructive to look at
those entities that can be both perceptually and intellectually pleasing, i.e. are considered as
mathematically beautiful and simultaneously have an attractive visual presentation, to see
whether mathematicians differentiate the two types of beauty. In fact they do!
Using a case study from graph theory (the highly symmetric Petersen graph), this talk tries to
distinguish genuine aesthetic from epistemic or practical judgements, and correct uses of the
word “beautiful” from loose ones. It demonstrates that mathematicians may respond to a
combination of perceptual properties of visual representations and mathematical properties of
abstract structures; the latter seem to carry greater weight. Mathematical beauty thus
primarily involves mathematicians' sensitivity to aesthetics of the abstract.
The next move would be to return to mathematical entities which are both epistemically
functional and regarded as mathematically beautiful, to see if mathematicians also differentiate
the epistemic value from the aesthetic value in these cases.