Not a lot of people understand or appreciate mathematics. Most of the people I grew up with shudder whenever mathematics is mentioned. “I’m not good with numbers” or “I never was a math person” are the two quintessential statements that non-math people utter whenever they’re confronted with mathematics.

Mathematics is so much more than numbers and numerical manipulation. Through Calculus 3 and somewhat into Differential Equations, numbers play a large role in evaluating the concepts and the fundamentals. It isn’t until the more advanced topics like abstract and linear algebra, topology, real analysis, and number theory that students actually develop an appreciation for what the UF Math department calls “Pure Mathematics.”

What is pure mathematics? Pure mathematics is the branch of mathematics that concerns itself with rigorous proofs and in-depth logical analysis of mathematical phenomena. Pure Mathematics is what Newton, Leibniz, Kepler, Gauss, Euler, and Godel studied. The application of logic (and sometimes philosopy) to understand the nature of the universe - that which surrounds us.

Physicists are notorious for being the ones that apply mathematical concepts to derive fundamental physical laws from the results of mathematics, but the rules are rarely bounded by reality. Mathematics often proves results which can never be realized or visualized in terms that humans can readily understand. Take, for example, a project I embarked upon while I took Abstract Algebra from Dr. Neal White. I wrote a C# program to solve a 3×3x3×3 Rubix Cube. That’s right..a 4th dimensional puzzle. When we look at a traditional 3×3x3 Rubix Cube, we are generally interested in the operations called “rotations.” We can rotate the entries of the cube. Each operation results in a movement/reorientation of pieces. Whenever we look at the cube though, we look at attempting to solve each side of the Rubix Cube. We know that the cube is completed when every side of the cube is the same color. In the fourth dimensional case, we know that the hypercube is solved when each face (in this case each face is a cube instead of a 2 dimensional side) is the same color. Operations are higher order also and have lots of very peculiar effects. Playing with a 4th dimensional cube as I did when I was testing the program isn’t intuitive - it’s natural for it to not be intuitive because we aren’t 4th dimensional creatures. We’re used to manipulating objects in three dimensional spaces. As soon as we start playing with things for which we have no real intuition, we get very bad at predicting behavior.

Playing with a Rubix Cube does not involve numbers. Nor does it really involve super-sophisticated logic to understand. Yet a Rubix Cube represents a fairly simple cyclic group from abstract algebra.

So why do some people find mathematics so interesting while other people look upon mathematics with disdain (or worse as some mystical skill that only the geeks possess and the jocks ignore)? I think a lot of it has to do with the way mathematics is taught in schools. If a student is more interested in obtaining the correct numerical value for an integration operation instead of being interested in why the integration operation exists, how it relates to differentiation, why it relates to differentiation, and under what circumstances Riemannian integration works, then the system is doomed to failure. Mathematics shouldn’t be a course generally regarded as one where partial credit is the savior. “Oh it’s okay I think I got enough partial credit to pass.” Students shouldn’t feel discouraged because they missed the orientation of a vector and marked the result with the improper sign. Students should instead feel terrible when they forget to include the “dx” term in an integral. Students should feel terrible that when discovering an anti-derivative that they omitted the “+ C” where C is a constant. We need to emphasize less on the numerical results of mathematics and more on the philosophical results of mathematics.

I’d like to share one final story about mathematics that I will forever remember. I was studying Number Theory with Professor Chen, and the course grading structure was thus: 85% homework, 15% final exam. Dr. Shen would assign a series of 4 problems to be completed and turned in for assessment every two weeks. For the typical college student taking the course, they would regard the homework as “a joke” and procrastinate until the day before the assignment was due to begin work. However, in Number Theory, those students were sorely mistaken. After the first few assignments, we all realized that the course was not a joke. The ideas that Dr. Shen asked us to prove had already been proven. The Truth of those theorems had already been discovered. In many cases, I would run a proof to near completion only to discover that I had assumed a fundamentally flawed notion, and I had to refocus my efforts on another strategy to solve the theorem.

Days would pass where I would obsess about the theorem. The emotional effort I was expending reminds me of being in love with a girl I would never meet. She existed out there - the theorem. In the realm of mathematics, she was living her life and affecting the world around her. I would obsess about the proof for days. In many cases, I would dream of potential solutions or approaches to the proof and wake to discover that they were either partially correct or completely erroneous. Damn the stuff of dreams! I could not stop thinking about the proof. Every simple breakthrough was an excitement. I would catch a glimpse of the proper approach at times and become so excited that I would continue my investigations at work instead of doing the web development for which I earned my meager $13.25 / hr. As I slowly revealed more and more of the solution to myself, I became more and more excited. Ideas became to come together and unravel the path to prove the theorem. Like gently undressing a lover for the first time, I would be careful - sometimes the path I had chosen was incorrect and I needed to be sure before committing time to an approach. As the solution became clear, more results popped out. These corollaries were almost as exciting as the proof of the original concept itself.

And so, for me, higher level mathematics always seems erotic - undressing a theorem to see how it really operates and describes reality.