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So enough talking about mathematical proof. Let's get to it.
-- Okay, smart guy. How do we start?
Well, if you toured the hallway, you know that I think that all mathematics starts with asking a question. Now if you're a student, you will probably get assignments like "Prove that thus and such is true." So, early in your mathematical career, you might not have much choice about which questions get asked. The point of these assignments is to give you practice at writing proofs. That's how everybody starts, and the practice can get you used to some the techniques and forms. But even if nobody else mentions it, even if your homework assignment tries to shortcut the process, remember that it all starts with wanting to find the answer to a question.
--Where do the questions come from?
Well, if you keep your eyes open, and let your mathematical imagination loose, just look at what's going on around you, and start asking questions. For me, I usually start with a few specific examples and try to find patterns.
Here are a couple of simple, specific examples to get started with:
Okay, that's more than a couple, but I get carried away sometimes.
So, what pattern does this suggest? What do you notice?
There are lots of things to notice, but for now, one thing I notice is that it seems that an odd number plus an odd number equals an even number, an odd number plus an even number equals an odd number, and an even number plus an even number equals an even number.
| 3 + 5 = 8 3 + 6 = 9 2 + 4 = 6 |
(odd + odd = even) (odd + even = odd) (even + even = even) |
-- So that's all there is to it? That's a proof?
Not hardly. This is the seed out of which a proof can grow. But since you asked, let's stop for a minute to talk about some formal matters.
Suppose I tell you that every car out in the parking lot is red. That's my claim. It may be right, it may be wrong, we don't know yet. I have to PROVE my claim.
So, I take you out to the parking lot and point at my car, and it's red. Does that prove my claim?
I don't think so. I made a claim about ALL the cars in the parking lot. Showing one example, my red car, doesn't prove anything about the rest of the cars. The rest of the lot might be full of blue cars. All I have proved is that there is at least one red car in the lot. To prove my claim I have to account for every single car in the lot, not just one example.
However, you can disprove my claim by pointing at one blue car. The fancy mathematicians call this a counterexample which is the twenty-five cent word for "Nope. You're wrong."
The only way I could prove my claim that every car in the lot was red would be to look at every car in the lot. We could do that. We could take the time to walk through the parking lot and check every single car. When we were finished, we would know whether all the cars were red, or they weren't.
But suppose instead of cars in a parking lot, I make a claim about every even number. Could I prove my claim by looking at examples?
No, that's like pointing at my red car. Even if I point at 5 red cars, or 25 cars, I still haven't accounted for all the cars.
Can I prove a claim about every even number by testing every even number?
No, because there are too many even numbers. Even if I prove something by looking at the first billion even numbers, there are still infinitely many numbers I haven't tested yet. It's like a parking lot that goes on and on. I'll never be able to check every one.
-- Okay, that won't work. Now what am I supposed to do?
If you've seen proofs in your math books, it is important to remember that by the time the finished proof is published in the book, the authors had lots of time to make it perfect. They wrote it, and rewrote it, and sent it to their editors, who sent it to the proofreaders, and lots of people got a chance to polish their proof to make it the perfect gem that sparkles on the printed page. It probably didn't start off that way.
Mathematicians cover their tracks. I mean, they show you their perfect, polished, edited proof, but only rarely do they give you any hint of how many bad early versions went in the wastebasket. Throwing away early versions doesn't mean you're a bad mathematician. My experience is that frequently I don't really understand the claim, and what it will take to prove it, until I've tried to write it out. The early attempts, that I throw in the recycling bin, are a valuable and necessary (for me) part of the process.
If you ever look at the hand-written manuscripts of music by Mozart and Beethoven, you will notice something very striking. Mozart's manuscripts are almost perfect. It seems like before he picked up his pen, the whole composition was already written in his head, and all he had to do was to put it down on paper. On the other hand, Beethoven's manuscripts are full of passages crossed out, scribbled over, and rearranged. Beethoven had to see things written out in front of him before he could tell that something didn't work.
Here's another mathematical secret I want to share with you. Most of us are more like Beethoven than Mozart. Don't let the proofs in the math books put you down.
So, what I'm trying to say is when you first start to write out a proof, you should say to yourself "This is just the first draft. This is scratch paper." Expect to cross things out and throw some ideas away. Beethoven did. You're allowed.
Back to the proof.
This is the scratch paper.
I claim that the sum of any even number added to any even number is an even number. I can't prove this by trying every possible combination of even numbers. There are too many.
To prove my claim, I need to generalize the specific examples, somehow, so I can make mathematical statements about all the even numbers at once. So, looking at all the examples of even numbers, I need to abstract the feature that they all have in common.
So how do I make a mathematical statement about every even number? Well, frequently it helps to check the definition. It's tough to go wrong starting from a clear understanding of exactly what we're talking about. So what is an even number?
An even number is a number that is divisible by 2. You can divide an even number by 2 and not leave a remainder. Or, to put it another way, an even number is 2 times some whole number.
an even number = 2 times some whole number
This means that I can write the sum of 2 even numbers this way:
(an even number) plus (an even number) = (2 times some whole number) plus (2 times some other whole number)
Now that I can explain this in plain words, it's okay to introduce some mathematical symbols. I recommend that you always start off with ordinary words. If you can't state your idea in plain language, you won't be able to find the right mathematical symbols. But, once you do state your idea simply, it's much easier just to plug in some well-chosen symbols.
Suppose I let E and F be even numbers. Then I can find whole numbers, I'll call them M and N, so that E equals 2 times M and F equals 2 times N. There's nothing special about the letters. I could have chosen A, B, C, and D just as well. The letters are just shorthand. Instead of having to write out all the words to say an even number plus an even number equals 2 times some whole number plus 2 times some other whole number, I can rewrite the statement this way:
This is nothing new. I just substituted the symbols for the words they represent
All these symbols have worn me out. I'm going to rest for a minute. When you're ready, click on the link to wake me up and we'll continue.
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