Which is the most beautiful?

Abstract

According to Aristotle, "Those who assert that the mathematical sciences say nothing of the beautiful are in error. The chief forms of beauty are order, commensurability and precision" [1]. G. H. Hardy in a famous passage asserted: "A mathematician, like a painter or a poet, is a maker of patterns . . . . The mathematician's patterns, like the painter's or poet's, must be beautiful . . . . Beauty is the first test: there is no permanent place in the world for ugly mathematics" [2]. Von Neumann wrote: "I think it is correct to say that [the mathematician's] criteria of selection, and also those of success, are mainly aesthetical" [3]. Poincar6 reflected a similar opinion when he wrote, "It is true aesthetic feeling which all mathematicians recognise . . . . The useful combinations are precisely the most beautiful" [4]. Weyl claimed: "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful" [5]. Morris Kline emphasised a different perspective: "Much research for new proofs of theorems already correctly established is undertaken simply because the existing proofs have no aesthetic appeal" [6]. Beauty does seem to be an essential, if little discussed, aspect of mathematics and the work of mathematicians. Yet no one can say precisely of what beauty in mathematics consists, and professional mathematicians will not necessarily agree on their definitions of mathematical beauty, on their practical judgments of which theorems, proofs, concepts, or strategies are the most beautiful, or on the role their personal feelings for mathematical beauty play in their own work. This questionnaire is a simple attempt to gather some data on the preferences of readers of the Mathematical Intelligencer. The least you are asked to do is to give each of these theorems a score of 0 through 10 for beau ty . (The mos t beaut i ful theorems score the highest marks.) Notice that you are not asked to judge between diff e ren t p roofs of the same theorem, or b e t w e e n theorems and proofs. I appreciate that readers are bound to be influenced in their judgments by their knowledge, or lack of knowledge, of particular proofs. Given constraints of space, and readers' time, I decided to focus on 24 different and varied theorems, all relatively easy to unders tand as statements, rather than a small handful of deta i led proofs or o ther aspects of mathematical beauty. Meta-responses--for example, that this questionnaire is imposs ib le to answer , or that part icular theorems cannot be ranked with merely a n u m b e r are we lcome and will be cons idered as valid as straightforward rankings of the theorems. Additional comments of any kind will of course also be most welcome. I hope to report on readers' views in a future issue of the Mathematical Intelligencer.