Mathematical proofs can be established by various means, including induction, contradiction, construction, and exhaustion. My favourite is proof by intimidation.
Proof is a core part of mathematical practice. If you can find a proof for a mathematical theorem, that proof will last as long as the axioms of mathematics do. Prove something difficult or long unsolved, you become famous. The British mathematician Andrew Wiles found a proof for a theorem that had gone unsolved for more than three hundred years, and now he’s one of the best known mathematicians in the world. (You’ve heard of him, right? I’d give you a full biography, but this blog post is too narrow to contain it.)
Proof by induction, for example, has two steps. First, you prove a single case – for example, involving just one number. Second, you prove a rule that shows how it’s also true for that number plus one. Now that the second case is proved, the rule automatically proves the third, the third proves the fourth, the fourth proves the fifth, and so on. Now, technically that’s deduction and not induction, but they’re mathematicians not grammarians.
Proof by contradiction is a nice one: you demonstrate that, if something were true, it would result in a contradiction. You can prove that the square root of 2 is irrational – i.e. it cannot be represented as the ratio of two integers – by imagining that hypothetical ratio and then demonstrating that it’s mathematically impossible for it to exist. A persistent myth holds that the Greek philosopher Hippasus was drowned in the sea by the Pythagoreans for proving that, by the way.
Proof by intimidation is mathematician humour. It posits a proof that is so full of technical jargon and overwhelming complexity that you have to accept it, or you would effectively be admitting that you’re not good enough to follow its intricacies.
Okay, I’ll admit it, to me most proofs are intimidating.