The Beauty of Mathematics

Those who cannot appreciate the “cold and austere”¹ beauty of mathematics are looking at the package and not the contents. One finds in the packaging a world of strict, formal symbols and fixed (non-creative) rules for manipulation. Non-creative arenas by definition resist the ultimate human act of creation (read: art). But it’s not in the results that we find the principal beauty of mathematics–it’s in the creation, the problem-solving, which requires the greatest human creativity the world has ever seen. It’s a realm of dazzling breadth and infinite depth; a shelf of approaches to each of a lifetime of topics.

In the world of computer science it’s commonly noted that there’s no algorithm for the design of algorithms, which is to say that the solving of any given computational problem requires unique, creative effort. We can’t write a program that can write all of our other programs—the same carries over into mathematics.

If you believe that mathematics is routine manipulation of uninspired symbols, odds are you’ve studied so little math that you only understand the fixed routines which are the building blocks wielded unconsciously in the creative energies of the mathematically adept.

Why is math so disliked? Because it’s challenging. Many of us struggle with the nuances of the basic manipulations, and that means we never get to the creative part. Any child can pick up a brush and paint—good painting is hard but simple painting is easy. Simple math is still hard. You can’t readily intuit it.

But as you progress through the echelons of mathematics you find that the nature of math education changes markedly. Instructors are no longer so concerned about your numeric mistakes or accidental sign changes–the very things they focused on previously. Strangely enough, the numbers most of us think of as math start to disappear entirely. Patterns are reified and particulars are relinquished.

It becomes a world of grand relationships and universal truths. The possibility of changing the entire nature of truth waits around every corner of calculation. An unexpected answer could mean a mistake or could mean you’ve revolutionized the understanding of man.

I’ve long thought math educators² squandered their significance in reducing the manipulation of equations to an impertinent, mechanical process. Computation over epistemology³. The substrate is truth, not numbers! You don’t add 2 to both sides—you leverage the universal truth that abstract quantities (any in the universe!) held in equivalence will remain in equivalence when modified by constant addition. You’ve amalgamated a truth and a metatruth into a third, irrefutable, novel truth.

Math, rightly viewed, is a world of grand, democratic intrigue where a legend can be dethroned with the stroke of a pen. Even the dead are not safe. And every last student in every corner of the world holds this unbounded power in the palm of his or her hand. If that’s not romantic, I don’t know what is.