In his note to students of mathematics, Michael Atiyah advises:

“The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable.”

The term “interesting” is not free of subjectivity. A trained mathematician can certainly draw on their deep knowledge and experience to filter through problems, but determining whether a problem is worth your time also requires a personal judgement. If this is ‘good mathematics’, then the ‘bad mathematics’ of schooling disempowers students from posing their own questions, or to consider what makes a problem interesting or solvable.

In his famous Apology, G H Hardy professed plainly that

“I am interested in mathematics only as a creative art.”

This is an astonishing statement when you consider that Hardy was a pure mathematician. Hardy was fiercely attached to the formalities of rigour and proof. His manuscripts, replete with symbolic abstractions, do not immediately strike one as creative or artistic. Yet for Hardy, a mathematical argument, while bound to logic, had to be crafted with artistry.

This hardened thinker was not without feeling:

“Like creative art, maths promotes and sustains a lofty habit of mind, increases happiness of mathematicians and other people.”

Hardy found delight in mathematical reasoning. His towering intellect was matched by a simple understanding of a mathematician’s primary purpose:

“A mathematician, like a painter or a poet, is a maker of patterns.”

Hardy even laid down a criterion for what constitutes ‘serious’ mathematics:

“Beauty is the first test. There is no permanent place in the world for ugly mathematics.”

Most of school mathematics, with its rigid focus on tedious procedure, would fail Hardy’s test. Bertrand Russell, a contemporary of Hardy’s, defines mathematics in terms of its beauty:

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere…sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

Where Hardy apologizes, mathematician turned teacher Paul Lockhart laments. His two-part essay begins with a scathing attack on school mathematics:

“No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression.”

Lockhart too cannot escape the characterization of mathematics as art itself:

“Mathematics is the purest of the arts, as well as the most misunderstood…The mathematician’s art [is] asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations.”

We can quibble over the precise definition of art. But whatever it is that makes music, poetry and painting art forms must also apply to mathematics. Art lies in process, not outcome. We experience art with emotion and intellect. We have mental and physical — even spiritual — reactions to it. Mathematics, in its true form, elicts the very same. While mathematical truths are absolute, the manner in which we discover and engage with those truths is anything but.