The Beauty of Mathematics – A Rough Sketch for a Proof

Mathematics and the aesthetics enjoy a long history of mutual references, which may be explained by the fact that both domains are dealing with relations: structure, content, schemata and (dis)similarity (cf. Wechsler 1978). And beauty? Beauty is the experience of pleasure and amazement from something that is aesthetic. Although “theorems and proofs which are agreed upon to be beautiful are rare” (Rota 1997), many mathematicians describe the process and/or the results of their work as beautiful, giving them and others an aesthetic experience. The French universalist Jules Henri. Poincare (1854-1912) suggested that: “the mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.” In addition to numerous examples of beautiful images, pattern and visualisations in mathematics, or generated with the help of mathematics, there is the question whether aesthetic elements are a vital component in the process that generates mathematical results. Poincaré was one of the first mathematicians to draw attention to the aesthetic dimension of mathematical invention and creation. For him the aesthetic plays a major role in the subconscious operations in a mathematician’s mind (see Hadamard 1945). In this essay I am going beyond a general discussion of how beautiful mathematics is and instead I will try to pin down an aesthetic element in a proof, providing support for the view that that the distinguishing feature of the mathematical mind is not logical but aesthetic. My investigation is motivated by Nelson Goodman’s theory of symbols (1976). Although I was not able to map all of Goodman’s symptoms of the aesthetic (syntactic density, semantic density, syntactic repleteness and exemplification) onto the notion of proofs in a satisfactory way, I believe that the distinction between denotation and exemplification in the proof for the irrationality of 2 allows a more focussed discussion about the evaluative and generative role of the aesthetics in mathematics. Furthermore, I also suggest that the notion of exemplification in the sense of Goodman reveals ambiguity, which subsequently plays a positive role in the development of the proof (cf. Byers 2007).