Organization of papers

I'm currently attempting to organize a paper out of a bunch of notes I've built up recently; a possibly useful suggestion I received is to write each theorem, definition, etc. on an index card, so that I can physically move them around to figure out how the paper should be organized.

Of course, definitions have to come before the theorems that use them, some theorems use other theorems in their proofs, and so on -- so to the extent that I'm remembering to do so, I'm indicating these sorts of dependencies on the index cards as well.

It occurs to me that what I am doing here is trying to extend a partial order (the ordering that comes from the dependency) to a total order. There are of course constraints on this order; certain results, although not logically related, are related in some philosophical sense and should perhaps be kept near each other. It's actually an interesting optimization problem.

Now if only I were writing a paper about extending partial orders to total orders...

(But my paper does talk quite a bit about permutations. And a total order will end up being a permutation of my index cards.)

The Olympic poset

During the Olympics (yeah, I know, you've all forgotten about the Olympics), there was much argument about whether the right thing to do, if we want to determine which country "won the Olympics", is to count the country that got the most gold medals (China), the most medals (the US), or something else -- say a points system that allocates three points for a gold, two for a silver, and one for a bronze.

Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:

So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades.

This gives a partial order on the countries.

Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.

One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?