Modeling aesthetics in mathematics

What exactly is beautiful math?

[A]bove all, adepts [of mathematics] find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?

-Henri Poincaré, The Value of Science

One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects “elegance” in its “architectural”, structural makeup. Ease in stating the problem, great difficulty in getting hold of it and in all attempts at approaching it, then again some very surprising twist by which the approach, or some part of the approach, becomes easy, etc. Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which “explains” the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc. These criteria are clearly those of any creative art.

-John von Neumann, The Mathematician

The moral: a good proof is one that makes us wiser.

-Yuri Manin, A Course in Mathematical Logic for Mathematicians

My hypothesis is that generally when people talk about beauty in mathematics they’re talking about things that teach us something useful for proving new facts. For example, proving a difficult but simple theorem is useful because its difficulty means it may imply other previously difficult theorems, and its simplicity means it may show up and be used often. A theorem that establishes a connection between two previously disparate areas of mathematics is considered beautiful, and such a connection allows knowledge from one are to be applied to the other, potentially cracking new problems. An unexpected proof – “an unexpected perspective” or “surprising twist” – offers something new to be learned, something that can then be used for other problems.