Aesthetic Preferences in Mathematics: a Case Study
Abstract
Although mathematicians often use it, mathematical beauty is a philosophically chal-
lenging concept. How can abstract objects be evaluated as beautiful? Is this related to
the way we visualise them?
Using a case study from graph theory (the highly symmetric Petersen graph), this
paper tries to analyse aesthetic preferences in mathematical practice and to distinguish
genuine aesthetic from epistemic or practical judgements.
It argues that, in making aesthetic judgements, mathematicians may be responding
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.
Introduction
What is mathematical beauty? How could beauty be found in such an abstract subject
as mathematics? Not attempting to solve this problem, I will attempt to answer a pair
of specific questions which have some bearing on it. These questions arise from math-
ematicians’ judgements about both the abstract objects they investigate and the visual
artefacts they use to represent those objects. These seem to be aesthetic judgements,
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often using the word “beautiful”, about the objects and their representations. Could
such judgements, taken literally, be correct? Specifically:
(i) Can an abstract mathematical object be literally beautiful?
(ii) Can one of its visual representations be more beautiful than others?
The last question needs to be refined. A visual representation can be appreciated
without regard for what it is intended to represent, and as such could be beautiful in
a way that a non-mathematician could appreciate. But that is not what is meant. A
better way to put the question is this:
(ii) Can one visual representation of a mathematical object represent its object in
a more beautiful way than another visual representation represents the same object?
And what does this mean for a mathematician?
Mathematicians’ talk suggests a positive answer to both questions.
But the pos-
sibility of loose or metaphorical uses of aesthetic expressions entails that we should
be cautious about taking such talk at face value.
In what follows I will first very
briefly set out some of the positions that philosophers working in aesthetics have taken
related to these questions.
I will then introduce a particular mathematical example,
an abstract object and representations of it, and mathematicians’ apparently aesthetic
judgements about them. Finally, I will discuss these examples with a view to answering
my questions.
The ongoing discussion in aesthetics
There is an on-going discussion about whether there is such a thing as mathematical
beauty, since there seem to be two options to consider, namely that beauty, apart
from sharing a propensity to give pleasure of a certain disinterested kind (i.e.
non-
instrumental pleasure), is:
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1. only perceptual, i.e. dependent only on properties, which an object can be per-
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- Fall '19
