Spoilers in Mathematics
A good theorem is a good mystery
Mystery is an intellectual process, like in a whodunit. But suspense is essentially an emotional process. Therefore, you can only get the suspense element going by giving the audience information. — Alfred Hitchcock
The journey through a good mystery is nothing short of a suspenseful intellectual experience. After introducing the main characters and some back-story, the story develops and a conflict spawns. The mystery bubbles up, perhaps there is sign of blood, or a hint of lies, leading up the an enigmatic murder.
“What could be the cause?” I try to logically piece together each event. The game is in the deduction. I enumerate through all possibilities and formulate some of my most creative conjectures — “I think I know…”
A good theorem should read like a good novel. If you were to open any textbook on Mathematics today, you will be welcomed with an assortment of definitions, theorems, and proofs. You will certainly notice that theorems are almost always stated before their proofs. Or, in other words, movie spoilers are revealed before the plot. Undergraduate Mathematics is the worst movie ever. It is a backwards experience. We are told the answers, and then asked to prove them.
The process of Mathematics must also be one of suspense. Axioms are the core pieces that lay the foundations of truth. They are, so the speak, the first few minutes of the movie or novel consisting of character development and setting. With this collection of facts, we obtain the tools necessary to form astounding conclusions, filling up a bank of Mathematical knowledge. Each mathematical proof is like the story described above. The axioms are thrown together in a crucible to form truths that are again used to bootstrap more interesting conclusions until perhaps the main mystery is solved. And each conclusion made using just previous knowledge is called a theorem.
Mathematics deserves a revamp. The old-age pedagogical techniques have much catching up to do. How does one expect us to forge new branches of mathematics without first building an intuitive understanding, when the current state of education is much like fixed guard rails.
Working bottom-up using low-level axioms and lemmas simulates mathematical research. In the real world, we explore the unknown. Rarely are researches given a truth and told to prove it, akin to mathematical education today that emphasizes proving well-known theorems instead of developing insight into the intuition behind the theorem.
Teachers, professors, and educators:
Allow us to work out some of the proofs in-class by ourselves. Let us develop the intuition before we start the proof, and let the proof lead itself to the theorem. Let’s go over difficult homework problems together in class, and finish up easy-to-follow lecture notes at home. Can we have lecture videos posted online, or at least the lecture slides to follow along? Help us develop mathematical wisdom, not just mathematical knowledge.