The complementary faces of mathematical beauty

This article focuses on the writings of Hardy, Poincaré, Birkhoff, and Whitehead,
in order to substantiate the claim that mathematicians experience a mathematical
proof as beautiful when it offers a maximum of insight while demanding a minimum
of effort. In other words, it claims that the study of the aesthetic success of
theorem-proofs can benefit from the analogy with the economic success of a business,
which involves maximizing return on investment. On the other hand, the
article also draws on Le Lionnais and Whitehead (again) in order to show that,
whereas the kind of aesthetic delight offered by beautiful proofs is typical for wellestablished
branches of mathematics, a romantic and adventurous spirit that goes
beyond the search for classical aesthetic delights is needed when the exploration
of new mathematics is at stake. The history of mathematics is not only a story of
feelings of beauty invoked by perfect products, but also a survey of sublime periods
of creative production. No account of mathematical beauty can be complete if it
does not complement the classical product aesthetics with a romantic creation
aesthetics.