Some Thoughts on Proofs
Note: This was written for the students in T403 fall 1998.

A lot of people have been mentioning their trouble with proof-writing. Here's what I wrote in one person's journal. (Almost--I revised.)

There are two really hard things about writing proofs. One is that it's a proof, so it's supposed to be all formal and so forth. The other is the logical progression of the ideas. I always work on the second thing first. I often have conversations with myself in which I try to explain to myself why something is true. The self being explained to is always playing devil's advocate. Sound a little insane? Perhaps it is, but this is exactly what you do when you make any decision, such as whether to go to the movie today or tomorrow. You weigh your options. You argue with yourself. One of you wins, and then you go that day. So you are familiar with this process, just not in a mathematical setting.

Once you have convinced yourself of each point in your logical progression of ideas, you try to write down this progression. That becomes what you think of as the proof. But really this whole process is the proof. It's just that no one sees the first part.

One suggestion I have is: Don't hold back on the devil's advocate side. Don't ever just let something slide by. It is this that you do when you say something is just obvious, but what you really mean is that you can't think of the individual parts that make it true, even though you believe in your bones that the thing is true. It's very tempting to just accept something, but that is often where mistakes are made.

The part where you turn the logical progression of ideas into a written statement is hard. You have to write down each thing that you thought. Often it's hard to remember them all, and so you have to go through the argument again several times. Usually, too, you might write something and then decide that's not the best way to present it. This happens all of the time in class when I'm using someone else's suggestion for a proof, rather than the one I brought, all nicely polished, from home. Then we start one way, erase, and say something else first. Almost no one sits down and writes a perfect proof from beginning to end. Don't expect that from yourself. The tricky thing is that whenever you read a proof, it'll look like the author did just that. You don't see all of the times that the sentences were rearranged. You don't see what got erased or reworded. You have to revise.

Maybe realizing that the process is not straight-forward will help you understand that one of the important things about proof writing is perseverance.