What I’m saying is that proofs are bigger than the problems they come from. A proof tells you what really matters and what is mere fluff, or irrelevant detail; it separates the wheat from the chaff. Of course, some proofs are better than others in this regard. Often a new argument is discovered that shows that what was previously thought to be an important assumption is in fact unnecessary. I suppose what I’m really trying to say here is that mathematical structures are designed and built not so much by us, as by our proofs.
