Partial results

by Orr Shalit

What is more wonderful: something wonderful, or the moment before the wonderful something happens?

Recently I proved a result that can best be described as a partial result. This is certainly not what I planned to obtain. My goal was to prove C. The plan was as follows:

  1. Prove A
  2. Prove B
  3. Prove that A and B together imply C

Step 3 is easy and I had it from the start. I’ve been trying to prove A and B for a several months now. I was sure that step 1 is the easier part and that step 2 is harder. I was quite hoping that step 1 follows from a general theorem which I also wanted to prove, but would have been just as happy to find it in the books.

Last week somebody showed me a counter example to the general theorem that would have implied A. Two days later I proved B. Now it remains to prove A, but A is not going to be true for the reasons I thought it would. Have to try a different approach…

But wait! I don’t want to prove A yet (not that I believe that there is a real “danger” in that happening). I want to enjoy B. B is beautiful. If I prove A, then C follows, and B is a triviality compared to C.

Worse, if A proves to be impenetrable, then as far as C goes, B is useless.

The role of B as a step in the proof of C can be appreciated a little more if we wait. Even if A is never proved, I wish to take some time enjoy B.

Alright, time out over! Back to A.