Random House Webster’s (4th ed) defines art as “1. the production, expression, or realm of what is beautiful.” Traditionally, art is used to refer to paintings, sculptures, or other masterpieces of the visual arts. Art can also include performing arts, such as music and theatre.

Yet it has also been used in reference to many other disciplines. For example, Sun Tzu’s famous manual The Art of War is a treatise on complex military stratagems, ideas as intricate and finely crafted as any sculpture. Its visual beauty is surely lacking, but Sun Tzu’s vision is for a military campaign as masterful as any canvas. Indeed, this campaign is set to paint a picture: with tactics and small-unit actions as the individual brushstrokes, but a grand vision as wide and broad of that of any “traditional” artist. Another example is Vladimir Vukovic’s classic Art of Attack in Chess. A chess attack can be as subtle and multifaceted as any great work of art, and even the smallest misstep can turn a clear win into a dead loss. Vukovic teaches the reader how to creatively work the pieces in a delicate tango, in a fine balance, to ensure ultimate victory. Yet, just as much as any of these things, mathematics is the production of the beautiful, and thus is an art.

How is math an art? In most high school classes, math is anything but an art, most often focused primarily on dull computations and repetitive operations. Yet unlike this “plug-and-chug” style of math that usually involves simply plugging numbers into formulae, real math has strategy as complex as anything in Sun Tzu’s book. And like the choreographed ensemble of pieces working in harmony found in Vukovic’s book, high-level math has a profound beauty all to its own. Real math is not computation, but proof.

A mathematical formula or idea means nothing until proven. Generally, a mathematician has an idea for what he or she wants to achieve, and must conjure the end formula. Just as an artist must transform a canvas into his beautiful vision, a mathematician must turn the familiar properties of the tangible world, the real numbers, into a formula that seems as if it was from another world. These proofs require immense creativity. Although there are general principles just as in painting or in the chess attack – for example, proof by contradiction or mathematical induction – no two proofs are the same.

Math is entirely based on a set of axioms, which are things that we assume about our world. From these basic axioms, ideas as simple as 1 + 0 = 1, all of math can be constructed. Of course, the math that appears in our everyday lives, addition and multiplication and so on, follow from this. But that’s not all. There is so much more math that results – even the limitations of what can be discovered are unknown.

There is a group of seven famous conjectures (results that are suspected but not proven) many of which are very old, each of which has a million-dollar prize attached to its resolution. These are known as the Clay Millennium Problems. They have resisted attack by the best mathematicians in the world for hundreds of years and despite the sweat and tears of the greatest mathematical minds ever to roam the earth, they remain unresolved.

Some conjectures of this difficulty are relatively simple to explain. The famous Goldbach Conjecture states that every even number greater than two can be written as the sum of two prime numbers (a prime number is a number that cannot be expressed as the product of two other numbers). Although it is true for all numbers less than a billion, efforts to prove it for all numbers have stymied the world’s best mathematicians for over 250 years.

Another easily explainable mathematical result was stated by Pierre de Fermat in 1637. He claimed that the equation xn + yn = zn has no positive integer solutions for n ≥ 3. For over 350 years, this had the status of a conjecture, as the world’s best mathematical minds ran into a brick wall. However, within the last 20 years, it has advanced to theorem status. One Andrew Wiles had dreamt of proving this incredible result from an early age. Eventually, the work of Ken Ribet made a major step towards proving Fermat’s Last Theorem, linking it to another conjecture. Wiles realized that the opportunity for a final proof was at hand, and devoted years to finding it. Eventually he was successful, and detailed his proof in a paper over 100 pages long. This paper is as masterful as any of the works of Michelangelo or Raphael. Of course, it is in a different form; instead of a large canvas, the proof is a smaller and thicker pamphlet. Instead of brushstrokes of color, Wiles used mathematical symbols: in this case, operations from what is known as Galois Theory. To the lay observer, these may seem arcane and incomprehensible, but to the mathematician they are no less beautiful than brushstrokes, masterfully combined and commingled to produce a finished work.

Instead of the painting masters, such as da Vinci and Monet, or chess masters, such as Tal and Shirov, math has its own personalities: notably, Carl Friedrich Gauss and Leonhard Euler. Their names and artistic results litter the pages of mathematics textbooks in many fields, where it is impossible not to encounter a beautiful proof by one of these two.

Yet in high school mathematics we rarely are aware of the art of math. High school results do not descend deep enough to see this: they barely skim the surface. Sadly, most high school interpretations of math are purely formulaic. Thus, mathematics is thought of as a boring subject. Something is fundamentally wrong with this approach to teaching math. Even at my high school, Stuyvesant, one of the foremost mathematical schools in the country, math is treated by many teachers and students alike as dull. This truly reveals the extent to which our math education system is broken.

But regardless of the boring way it is taught in high school, mathematics at a certain level indeed can be beautiful. Once one reaches the point of proving instead of plugging, math is, I feel, the most interesting subject there is. To me, mathematics is most definitely an art.

Acknowledgements: I’d like to acknowledge Joseph Stern (now at Columbia) and Jim Cocoros, both my teachers and mentors at Stuyvesant, for inspiring me to see the light of math.