Mathematical Structures, useful or not, are invented and developed within a problem context, and derive their meaning from that context. Sometimes we want one plus one to equal zero (as in so called ‘mod 2′ arithmetic) and on the surface of a sphere the angles of a triangle add up to more than 180 degrees. These are not “facts” per se; everything is relative and relational.
A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light — It should refresh the spirit and illuminate the mind. And it should be charming.
Even the traditional way in which definitions are presented is a lie. In an effort to create an illusion of “clarity” before embarking on the typical cascade of propositions and theorems, a set of definitions are provided so that statements and their proofs can be made as succint as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or structural property.
There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract.