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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, 1
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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: 2
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1. only perceptual, i.e. dependent only on properties, which an object can be per-
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