What is NOT a proof?
What follows is a list of arguments that often seem like a good idea at the time, but are not proofs. I was going to say something about "if you are inexperienced" but realized that these fallacies can seduce us all.
1a) I tried very hard and failed to do this, therefore it can't be done.
1b) The following method doesn't work, therefore it can't be done.
2) I let n = 1, 2, 3, 4 and 5, and found a pattern. Therefore this pattern is true for all n
3) The theorem is true for the following example, therefore it is always true.
4) Look at this picture.
5) The hint said to show THIS, so we can assume THIS is true.
6) I am restating the problem, and therefore it is true. (“Theorem: 7068555·2121301-1 is prime. Proof: 7068555·2121301-1 is only divisible by itself and 1”)
The thing is – these things are good steps on the way to constructing REAL proofs! To show something is impossible, you DO gain insight by trying to do it and seeing what goes wrong. To gain insight for a general n, you DO gain insight by letting n = 1, 2, 3, 4 and 5. To show a theorem is true, you DO gain insight by constructing an example. If you are lucky enough to be in a field of mathematics where you can draw a picture that represents what is going on, you DO gain insight both by looking at the picture, and by figuring out how to draw it. If you have a hint, you DO serve yourself well by taking it very seriously. And sometimes you DO find things easier to think about if you state them in a different way.
And most frustrated teachers, myself often included, WILL get excited and give you praise if they see you doing these things. And most teachers, myself often included, WILL do these things on the board.
So I understand how you may start thinking that the techniques above are proofs. But they are not. I wish I could make them so.
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Copyright 2012 by Douglas Shaw