This spring I will be teaching MATH 2001: Introduction to Discrete Mathematics. This is our introduction to the major here at the University of Colorado, Boulder, and most students find it a wild ride after their experience in Calculus.
What do I want out of this course? I want to impart to my students the experience and skills of a research mathematician. This may seem like the wrong goal. After all, our majors are mostly headed to other careers. Is it not a little bit of hubris, near-sightedness, or perhaps even isolation-induced narcissism, that pushes me to evangelize about my profession?
I prefer to think otherwise, but there is probably some truth to that assessment of my motivations. However, less-than-laudable motivations does not a bad goal make. So the purpose of this post is to explain what is valuable about the skill set I aim to teach in Math 2001.
First and foremost, my goal is precision in thought and communication. My goal is to harness and control fuzzy, associative thinking. Associative thinking begets creativity and intuition, two extremely valuable things in mathematics, but it must always be tempered by precision and logic, or it brings us only daydreams. I find that students usually arrive in 2001 with great creativity, but less ability to harness their creative urges to align with reality, mathematically speaking.
Phrased this way, one might expect this task to be somewhat depressing. But there’s another way to view the dynamic, empowering symbiosis between creativity and precision. Mathematics is not unlike art, in a very fundamental way. Reflect for a moment on the sonnet, a very rigid poetry form, with prescribed metre, rhyme and structure. One has a fixed total number of syllables to work with, and it is designated which of these may be given emphasis and which may not. The first eight lines set up or describe a problem; the ninth line is a turn in tone or mood. It is very demanding. And yet, there are an astronomically large (although finite) number of sonnets that could be written. And more importantly, struggling to match the form actually drives the creative process. It is actually much more challenging to write beautiful free verse. Mathematics is much the same. The mathematical artist has total freedom of subject, but she must hew to the rules of logic, like the rules of sonnet form.
This symbiosis is hardly unique to mathematics. An engineer must balance vision with the physical laws of his building materials. A scientist must balance the resources of his laboratory with the size of his motivating questions. But for some reason few people expect it in mathematics.
Perhaps this is because few people realize what a research mathematician does. A mathematician proves theorems. Theorems are logical inferences: if A is true, then B is true. To even imagine the statement of a theorem is a creative act, often born of long experimentation and exploration. The creative spark of a theorem statement may be presented by natural phenomena in our world, or by computational experiment. To prove the theorem also requires great creativity, in the form of logical arguments. The logic isn’t creative itself: it is simply a set of rules, like sonnet form. The creativity is in choosing which of infinitely many possible logical inferences is the right one. (We are a long way now from the recipe-like solutions taught in most high school calculus texts!)
Theorems are, like the bridges built by engineers, actually useful. They tell us how to program the scheduling of the rooms at the university, and guarantee the optimality of the outcome. They tell us how to successfully plan a trajectory to mars, and how to statistically support or refute the dangers of smoking or cell phones. The ability to create correct logical inferences — theorems — is at the heart of many of the jobs our mathematics majors will have in industry and elsewhere. To correctly design software to run a medical scanner, to decipher the scientific literature and make recommendations for treatment, to analyse cosmic radiation for signs of alien life, for any of the many jobs our majors have in their futures, one needs to think with precision and creativity.
My goal is to impart exactly those skills.
Postscript: I highly recommend the wonderful book How Not to be Wrong, by Jordan Ellenberg. This is aimed at a popular audience, and goes a long way to explaining how precise mathematical thinking can be useful in every sphere of life.