The mathematician as an artist

Mathematics represents the highest form of art. To those of you who perceive mathematics as simply the manipulation of numbers, such a statement seems, at best, a wild exaggeration. To justify this statement we need to consider what we mean by art.

The American Heritage dictionary (3rd edition) defines an artist (meaning 2) as "A person whose work shows exceptional creative ability or skill." Few would dispute the idea that mathematicians need exceptional skill, but many who do not practice mathematics are surprised to learn that creativity is the most important talent for a mathematician.

There is a notion that creativity means coming up with weird, shocking, but somehow pleasing things. Creative work can indeed seem weird, shocking and pleasing, but it is essential that an underlying concept lie behind it, giving it unity and purpose. Third rate artists can concoct weirdness and shock, but they lack the talent to make their work profound. That is what differentiates pseudo-art from true art. The art lies in having work transcend the strangeness and shock to give us something of beauty or understanding.

A mathematician is an artist in the sense that he or she creates structure from a set of concepts that he or she defines. The mathematician must define a concept in such a way that allows the construction of a structure that clarifies the relationships that exist between the elements. Every mathematical entity, from the most mundane to the most esoteric, is in reality a unifying concept. Take the number three. This is not something you can hold or feel in the same sense that you can a book, but it does describe a unity about a collection of objects or even concepts. Similarly, a mathematical group, describes a structure imposed on a set and an operation.

A mathematical proof can be beautiful in the same sense a painting is beautiful because it reveals heretofore unsuspected relationships between objects that are themselves concepts. It is the ability to conceive of new concepts that allow the structure to be revealed. For this reason mathematics represents the most exalted expression of human creativity