The Shortest Path
…between two points may be a straight line. But that’s not the shape of the usual path from a conjecture to a proof – whatever the writeup in our submitted paper may suggest. When we write a research paper, we leave out a lot. And that’s usually a good thing.
First, many of your readers will be busy researchers who can spare fifteen minutes to skim your papers for the highlights. Some will be driven by curiosity, some have just popped by to borrow a screwdriver to use in their own work, but in either case, the less they have to wade through, the happier they will be. Sure, they will want to know that the proof is there, but they’ll trust you and the referee for the rest.
Some will be more interested, or will want to see if your proof provides some techniques that they can borrow. While they’re going to read your proof, they’d still rather that you got to the point quickly. Proofs, after all, are heavy going.
So a research paper probably won’t say much about the experiments you did to get your conjecture in the first place. You mention a result that somebody proved fifty years ago, but you leave out the afternoon you spent proving it another way, just to see if it could be done. You tried five different versions of your main theorem, all rather similar, and you only include the two most interesting (and the proof of one.) Your final definition is much more elegant than the one you began with, and so that’s the definition that you keep.
And that’s as it should be. The results are for your colleagues, for posterity. But the journey you took to get there? That was for you.