What a Proof Is
The word "proof" makes a lot of people freeze - it sounds like a wall of Greek symbols only mathematicians are allowed behind. It isn't. A proof is something you already do informally every time you convince someone of a conclusion they can't wriggle out of: you start from things they already accept, take steps they can't object to, and arrive somewhere they now have to agree with. That's the whole idea. A proof is an argument with the gaps removed.
This guide demystifies it. You'll see what a proof actually is (and how it differs from "evidence"), meet the handful of standard techniques that cover almost everything - direct proof, contradiction, contrapositive, cases, the single counterexample - and then unlock the one that feels like magic until it doesn't: induction, which lets a few lines prove something true for infinitely many cases at once. If you've written a recursive function, you already think the way induction works.
How to read this
- Curious what proof even means? Phase 1 is the reframe.
- Want the toolkit? Read in order - Phase 2 is the techniques, Phase 3 is induction.
The phases
- What a Proof Actually Is - a gap-free chain from accepted truths to a conclusion, and why that's stronger than evidence.
- The Main Proof Techniques - direct, contradiction, contrapositive, cases, and disproof by counterexample.
- Proof by Induction - the domino principle, and why it's the same shape as recursion.
This builds on validity from What Logic Actually Is and the contrapositive from Propositional Logic. Last in the Logic foundations: spotting fallacies.