Introduction to Special Issue: Aesthetics in Mathematics†

Abstract

Mathematicians often appreciate the beauty and elegance of particular theorems, proofs, and definitions, attaching importance not only to the truth but also to the aesthetic merit of their work. As Henri Poincaré (1930: 59) put it, mathematical beauty is a 'real aesthetic feeling that all true mathematicians recognise'. Others went further, regarding mathematical beauty as a key motivation driving the formulation of mathematical proofs and even as a criterion for choosing one proof over another. As Hermann Weyl famously and provocatively declared, 'My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful' (cited in Chandrasekhar 1987: 52). Talk of the beauty of mathematical theorems, proofs, and definitions may thus be commonplace. And yet the tendency among mathematicians to judge mathematical work according to aesthetic standards raises a number of difficult questions: (1) What is mathematical beauty? What, if anything, distinguishes it from other kinds of beauty? Is it a feature of abstract objects or grounded in sensible properties? Is it a genuine aesthetic category or can it be reduced to non-aesthetic, possibly epistemic, criteria? (2) What is the status of aesthetic judgments in mathematics? Are they objective judgments grounded, for instance, in the mathematician's cognition of such properties as symmetry or simplicity? Do they rely on subjective responses