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.)
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4 comments:
When I was working on my book, I tried to explain this same point in almost exactly the same terms to my editor. I don't think I was very successful.
I often think of the problem in terms of finding a Hamiltonian path in a partially-ordered graph. The nodes of the graph are the topics I need to discuss, and the path is constrained partly by adjacency (which topics are related enough that there are good transitions between them) and by the need to extend the partial order to a total order (which topics are prerequisites for which).
Since the nodes in any one subject area will tend to form a highly-connected subgraph, a good heuristic is to consider the reduced graph that replaces each subject area with a single node, solve the problem first on the reduced graph, and then extending the path on the reduced graph by finding an appropriate path into and out of each subject-area clique.
This really is how I think about it, although I don't actually draw the graph. But I am often aware, as I write a book chapter or talk, of the backtracking search as I first find an approximate path that hits almost all the vertices, and then perturb it locally to shoehorn in the last few topics.
Let's just hope it IS a partial order--you don't want to lay out all your topics into a graph and find a cycle in there!
Nabokov wrote his books this way. One interesting side affect (supposedly, I've never checked it myself) is that the lengths of his pargraphs are bimodal: one index card or two.
I don't know if the order in which one writes a paper should always follow the logical ordering of the proof. For example, you should state your big theorems at the beginning so the reader knows what the paper is about, and it is often useful to state a major lemma before its supporting lemmas in order to motivate all the proofs of the smaller lemmas. The order of the proofs should often follow the logical order, but not always. If there's a long proof of a technical lemma that doesn't help with one's intuition, you may as well put it at the end of the paper.
I think a paper should follow an order that best motivates the discussion and promotes a nice flow of ideas, while keeping close enough to the logical order that the reader can piece together the logical flow on their own. But, of course, my research is in a subject where the types of proofs are probably quite different than yours.
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