An introduction to mathematical theorems - Scott Kennedy

What is proof? And why is it so important in mathematics? Proofs provide a solid foundation for mathematicians, logicians, statisticians, economists, architects, engineers, and many others to build and test their theories on. And they're just plain awesome!

Let me start at the beginning. I'll introduce you to a fellow named Euclid. As in, "here's looking at you, Clid." He lived in Greece about 2,300 years ago, and he's considered by many to be the father of geometry. So if you've been wondering where to send your geometry fan mail, Euclid of Alexandria is the guy to thank for proofs.

Euclid is not really known for inventing or discovering a lot of mathematics, but he revolutionized the way in which it is written, presented, and thought about. Euclid set out to formalize mathematics by establishing the rules of the game. These rules of the game are called axioms. Once you have the rules, Euclid says you have to use them to prove what you think is true. If you can't, then your theorem or idea might be false. And if your theorem is false, then any theorems that come after it and use it might be false too. Like how one misplaced beam can bring down the whole house.

So that's all that proofs are: using well-established rules to prove beyond a doubt that some theorem is true. Then you use those theorems like blocks to build mathematics. Let's check out an example. Say I want to prove that these two triangles are the same size and shape. In other words, they are congruent. Well, one way to do that is to write a proof that shows that all three sides of one triangle are congruent to all three sides of the other triangle.

So how do we prove it? First, I'll write down what we know. We know that point M is the midpoint of AB. We also know that sides AC and BC are already congruent. Now let's see. What does the midpoint tell us? Luckily, I know the definition of midpoint. It is basically the point in the middle. What this means is that AM and BM are the same length since M is the exact middle of AB. In other words, the bottom side of each of our triangles are congruent. I'll put that as step two.

Great! So far I have two pairs of sides that are congruent. The last one is easy. The third side of the left triangle is CM, and the third side of the right triangle is - well, also CM. They share the same side. Of course, it's congruent to itself! This is called the reflexive property. Everything is congruent to itself. I'll put this as step three.

Ta dah! You've just proven that all three sides of the left triangle are congruent to all three sides of the right triangle. Plus, the two triangles are congruent because of the side-side-side congruence theorem for triangles. When finished with a proof, I like to do what Euclid did. He marked the end of a proof with the letters QED. It's Latin for "quod erat demonstrandum," which translates literally to "what was to be proven." But I just think of it as "look what I just did!"

I can hear what you're thinking: why should I study proofs? One reason is that they could allow you to win any argument. Abraham Lincoln, one of our nation's greatest leaders of all time, used to keep a copy of Euclid's Elements on his bedside table to keep his mind in shape. Another reason is you can make a million dollars. You heard me. One million dollars. That's the price that the Clay Mathematics Institute in Massachusetts is willing to pay anyone who proves one of the many unproven theories that it calls "the millennium problems." A couple of these have been solved in the 90s and 2000s.

But beyond money and arguments, proofs are everywhere. They underlie architecture, art, computer programming, and internet security. If no one understood or could generate a proof, we could not advance these essential parts of our world. Finally, we all know that the proof is in the pudding. And pudding is delicious. QED.