|
|
In the basement, we look at the foundation of the house - mathematical proof. Wait a minute, before you go running back upstairs screaming, give me a chance. This won't hurt. Not much, anyway.
There are some people who define mathematicians as people who write proofs all day. I don't know about you, but that sounds like a drag to me. But, and this is a big but, writing proofs is a large part of how mathematicians record their results. Andrew Wiles made the evening news because he wrote a proof of a centuries-old mathematical guess.
So, every once in a while, after playing inventor for a while, mathematicians have to stop and write down their results, and their results are usually in the form of a proof.
Have you learned about the explorers yet? Human history is full of stories of exploration, of men and women who looked at the map of what was known, and then they went out there to the edge, and then they went a little further. They pulled up the fence post that was hemming them in, and pushed it out a ways. All of us who come after them can enjoy the benefits of their pushing against the boundaries.
Writing proofs is the way mathematicians pull up the stakes that mark the limits of what is known, and push them out into new territory. So, as you sit down to write a proof, you have company. You are taking your place in the great humankind parade of explorers, and Archimedes, Isaac Newton, Amelia Earhart, and Marie Curie are looking over your shoulders saying "Go ahead. Go for it. Push it a little. Let's find out what's on the other side."
A proof is a series of statements that can convince a skeptic of their truth. A proof is logically unassailable. There is no wiggle room in a proof. A proof that is not absolutely correct is not a proof.
You may have seen a courtroom scene on TV, where the lawyers talk about proving some claim "beyond a reasonable doubt". Well, that's about the most we can expect in many of our day to day activities. Experimental physicists and chemists and biologists and medical doctors repeat the same experiment over and over again, and get different results each time. It is very difficult to take precise measurements and get them absolutely correct. There is always a little error, so the scientists do lots of repetitions, in the hope that the little errors cancel each other out, and they hope they end up with something "reasonably close" to the truth.
That's not how mathematics works. Mathematics is one of the few human activities that can claim absolute certainty. This has its good side and its bad side. The good news is that it means once something is proved, then it's a fact, and you can use it, and everyone else can use it. Mathematics has a cumulative history. We can see farther today because we can sit on the shoulders of giants.
If you look up the word "prove" in a good dictionary, it will usually get around to something like "to determine the genuineness or quality of; to test," and that is the spirit you should have when you approach mathematics. A proof is a test, a contest against ignorance and the unknown. Every time you write a good proof, you have pushed the forces of ignorance a little farther away for all of us.
I'm glad you asked that question. But, before we dive into that, there are 3 pictures I want you to hold in your heads as we continue this investigation:
Image number 1.
Imagine a bridge, say, for example, the Golden Gate Bridge. What is a bridge? It's a way to get from point A to point B with dry feet. So you're standing there in San Francisco, and you just heard about a terrific new pizza emporium in Sausalito, but you can't swim, and you don't have a boat. So how are you going to get to the new tofu and sprout pizza? How are you going to get to the pizzeria? You need a bridge.
That's what a proof is, a bridge that gets you from over here (what you already know) to over there (what you think might be true). Writing a proof can be a lot of work, like building a bridge, but it's worth the effort when there's great pizza, or great math, on the other side.
Image number 2. Now imagine an arch, say, for example one of the arches holding up the Cathedral of Notre Dame in Paris. (I know, I'm a lousy artist, but I'm a decent mathematician.)
Arches are incredible architectural structures. They direct the weight from the building above them down their sides. With arches channeling the weight of the building, instead of having to have solid walls, you can put in windows, and let in a little light.
The stone at the very top of an arch is given a special name. It's called the keystone and it's usually the last of the pieces of the arch put in place.
I have found that when I'm working on proofs, there is usually a "keystone" idea, one essential piece of mathematics that I need to put into exactly the right spot. Once I find the keystone, most of the hard work is done, and the rest is the gruntwork of arranging the rest of the stones around it.
But make no mistake about this image. You need ALL the stones when you're building an arch, and you need all the facts when you're building a proof. You omit just one little stone, and the whole arch collapses. Just so with a proof. You can do all sorts of hard work, and get almost everything lined up, but if you're missing one little fact, the whole proof collapses.
Gravity is unforgiving, and so is proof writing. There is no room for "sort of" or "close enough". That may be why proofs are so intimidating for many students. Well, stick around, and I'll share all the helpful hints I have found so far.
But it also may help to think about this. The next time you're zooming skyward in an elevator, do you want the cables to be strong enough, absolutely-no-baloney strong enough. Or is okay with you if the cables are "almost" strong enough or "sort of" strong enough?
When I'm on the elevator, I don't want any "almost" or "sort of" strong enough. Almost strong enough means not strong enough.
Image number 3. Finally, picture a stream, with some stepping stones. Some of the stones are broad and flat, and solid, and some are wobbly and slippery, some are close together, and some will take a great leap to get to. You're trying to get from point A to point B, and you play connect the dots with the stepping stones, trying to see a path all the way across.
When you're writing a mathematical proof, you're trying to get from point A to point B, and the stepping stones are all the facts you already have proved. Once something is proved, it becomes a stepping stone in your stream, and you can use it to step on and go further. You just need to connect those facts so that there is a clear, complete path to the other side.
Sometimes you don't know the whole way across when you start. That's okay, I think. If you take the first step, you're already closer to the other side, so your problem just got smaller. If you do run into a dead end, and can't find any more stepping stones, you just back up a ways and try something different.
Even Andrew Wiles made mistakes. After he had made his proof public, he found a gap, a place where he had failed to connect the dots. So he went back to drawing board for several more months and fixed it.
Mathematics is not a spectator sport. To learn about proofs, you need to sit down and confront the unknown, and bring order to a little piece of chaos. But it's easier if someone is there as a guide. So come along with Spud, and invent some mathematics.
| Section 1 | Odds and evens - beginning |
| Section 2 | Odds and evens - middle |
| Section 3 | Odds and evens - end |