Crazy proof

My thesis is centered on looking at the history of a conjecture which could probably be considered a mathematical philosophy question.  It took nearly 30 years for it to be answered.  At the heart of the question - is the natural ordering system we have always give us the best result?  By natural ordering, I mean the basic way of counting - how we learn to organize and number - 1 cookie, 2 cookies, and so on.  I guess when I say "result" I am being a bit vague - I don't want to get into the nitty gritty here - I think I might lose everyone if I actually explained it.  Probably better think of it - suppose you have a very long list of grocery items and are looking to see if "carrot" is on the list.  What is the best way to go searching for it?  Start at the first item on the list and check them off?  Maybe start at the bottom of the list and work your way up?  What about start in the middle, consider the first half and check the middle of that, and so on and so on?  Any one of these methods could be considered an ordering system - which item comes first, second, third in each method?

What is interesting is that in some situations (which is a really good thing for scientists or well certain scientists) that we have found orderings that give great results (under certain conditions).  One group of functions has great behavior if we don't count by 1's but by powers of 2.  In the case of functions that relate to this conjecture, we find that the functions as a group don't behave as nicely as we want them to do, but we can do no better than this "ugly" nature if we try to change the ordering.

So you now you are asking me - well why did you name this entry "Crazy proof"?  The group of mathematicians who produced the proof submitted an article - the article is approximately 6 pages.  The proof of what is the major find, from which you can prove the conjecture in less than 3 lines, is about 2.5 pages long (which by many people's standards is unbelievably short).  I've spent almost 10 hours with my advisor working through the proof, plus another 10 or 15 trying to convince myself that things we spoke about were true and figuring out ways to spell all this out.  So my full complete version of the proof is the total length of original article.  Talk about taking some mathematical freedom when you get to submit journal articles.  Yet I think the proof is unsatisfying - there doesn't seem to be any explanation of how the authors came up with their idea - it does not allow any wiggle room for using the ideas in similar contexts.  It's quite possible the methods will lead to dead-ends if I try to use them in a new situation.  Oh well.

In the meantime, actual writing - 25 pages (it may seem a bit low compared to before - I had to rewrite several pages and combined several proofs), total page count - 29 pages.  I really only have about 2 sections left to work one - one that is more a historical account, the other my  work.  Oh yes - the introduction as well.  It's pitiful really.  I wish I had some ideas on how to rework that.

Lots of other craziness going on - the wedding detail stuff seems to be ramping up - at least a huge chunk of this work is out of the way.  Catch you all later.