I have often been sitting there swearing when I couldn’t figure out to how to prove something and I know for certain I am not the only one. But why are proofs so difficult when they seem so easy when you read them? This post is about the softer side of proofs, to encourage you to not loose faith when you meet something you can’t prove.
I think the answer to the question lies in the fact that what we see when other presents a proof is the result. It is completely void of any wrong turns and blind alleys. That doesn’t mean the person constructing the proof didn’t make them, but there is no reason to leave it in the final version, since it doesn’t help in the argument that the statement is valid. What we have looked at on this blog so far are pretty simple things where it was obvious where was obvious what method we should use to prove it. Mainly since I chose things that were well suited for whatever I wanted to say something about.
That is not always the case when tackling a problem which is more difficult. So I believe part of reading a proof is to ask yourself this question “Why did he he choose this argument?” and “Could he have done it differently?”. That helps me to unravel the thought process behind the cold proof and helps learn more about the statement than the fact that it is true. Admittedly I have read proofs which took turns and were so clever that I could never even begin to understand how the author had thought of that.
Many things we encounter especially during high school and such are proofs that were constructed thousands of years ago. I am sure Pythagoras and his fellows were frustrated as well, and the first time they managed to prove it the proof wasn’t nearly as nice as it is today.
I remember when I went to the Danish equivalent of high school. We saw a proof of the quadratic formula. I think my teacher wanted us to see that it wasn’t just magic, but it actually could be derived. At some point he comes up with the comment “And then we multiply both sides by 4”. I wasn’t self confident enough to question his authority but another student asked “Why do you do that?”, and the answer came “Because it gives us a bonus in the end”.
He was right, it did give some much nicer formulas, no doubt about it, but how or why he came to the conclusion that it was a good idea was never explained to us. I still believe that unless you had been through the whole proof you couldn’t have seen something like that. But going back and polishing it really does help with the whole argument and making it clear. And of course it is valid that we can multiply both sides by four. It just wasn’t obvious why we should do it.
A whole lot of text this time, about why you shouldn’t be frustrated when trying to do proofs. Since the process is different than the result. So hang in there and keep trying even if it seems that everyone else is being better than you.
Next time, lets try to do a proof and let’s see if we can get the thought process into it.
The top image for this post is taken by Marvin Lee and shared under the creative commons license.