by Brandon Summers | March 14, 2016
What most of us know as math is very superficial. I mean this sincerely and not dismissively. From elementary school, to middle school, to high school, to college algebra; from times tables, to long division, to factoring, to the pythagorean theorem, to the quadratic formula, to the interest rate formula, to functions, to derivatives, etc. we are taught how to do computations. We are shown a formula, the instructor shows us a few examples on the whiteboard, and then we are supposed to emulate that process– throw in some word problems for good measure, rinse and repeat. This is how we demonstrate that we know how to do math. However, this is not proof that we know why or how those formulas work. Advanced math courses address this by concentrating on the underlying structures of mathematics. Mathematical reasoning is at the core. There’s a good chance you have not done proofs.
…And that’s okay for 99.99% of students. Understanding a formula and knowing how to apply it is a useful skill that serves all of us well. However, it is not a skill that is particularly useful in advanced mathematics (i.e math post Calculus III). Mathematical reasoning/logic is the next step in growth (and it’s much different than K-12 math). Mathematicians, physicists, and some computer scientists need to have this skill in their toolbox. (Engineers, not so much. I would have avoided this had I majored in engineering). We take the course Discrete Mathematics as an introduction into the world of proofs. It’s a fundamentals course but it can be challenging for many. It’s a departure from plug-n-chug math. Calculators are not required (they’re essentially useless), and there are no formulas (hell, there are barely numbers)– only theorems and definitions. We are no longer solving for x. The answer is not a number, it’s the process. Here’s an example:
Prove that the sum of two even numbers is even.
Proof: Suppose x and y are even numbers.
Let x=2l and y=2m, for arbitrary integers l and m | (By definition of even numbers)
Then x+y = 2l+2m = 2(l+m)
Hence x+y is even since (l+m) is an integer. Therefore the sum of any two even numbers is even. (done)
Showing examples, say 4+6 = 10 or 12 + 12 = 24, is not sufficient. The goal is to prove that the sum of two numbers is even for ALL even numbers. That’s what proofs are about. You have to prove that assumptions hold true for ENTIRE sets. Many students struggle with proofs.
Note: This was a really easy example. Most proofs are A LOT tougher than that.
As math majors move beyond Discrete Mathematics, we encounter difficult/rigorous courses with increasingly difficult proofs. Not having a command of proofs is a death sentence. That was my fate. My stellar performance in Calculus II and Calculus III had no bearing on my outcome in Theory of Numbers, Abstract Algebra, Advanced Calculus, and Linear Algebra II. While I can “follow” a proof when I see one, I have always struggled with constructing proofs in regards to unfamiliar and complex problems. There are a lot of ways to go about proving things (direct, contraposition, contradiction, induction) and there’s a certain amount of “cleverness” that is involved in doing so. I lack the talent and the enthusiasm.
Why do proofs matter?
Math theory is the essence and foundation of mathematics. Theory has everything to do with why we (arbitrarily) count to ten, why we have prime numbers, how we have a pythagorean theorem, a quadratic formula, the number line, geometry, algebra, calculus, etc. Without mathematicians and their proofs, we wouldn’t have computers, cell phones, airplanes. I have nothing but respect and admiration for mathematical proofs; and I envy those who get it, but it’s not my thing.
The first homework problem in my Advanced Matrix Theory class:
1.1 Let A be an mxm idempotent matrix. Show that Im-A is idempotent.
Let A be an mxm idempotent matrix.
Then (Im-A)(Im-A) = (Im-AA)(Im-A)
= Im Im– ImA – AAIm + (AA)A
=Im -A + AA + AA
=Im -A -A +A