### Apprenticeship

Yesterday, I also got to watch my tutor figure out how to prove something that he had, perhaps, been shown long ago, but had long since forgotten. Proofs have often puzzled me: given a statement like "Prove that x is a Y if z(x) is prime" (I'm making that up but it's the right flavor) I would have no idea where to start.

What I saw yesterday is you really don't have to have any idea how the proof itself will start: you start by just writing down everything you know about the problem, and then begin "messing around" with the equations you have. Sooner or later, you (hopefully) will see the "finish line," as my tutor put it, and you will have your proof. Then you can clean things up, and get the concise, knee proves one finds published.

No book has ever indicated to me that proofs are done that way! And this is a very Oakeshottian point (or Polanyian, if you like your case for apprenticeship from a scientist).

1. I find it more helpful to work from both ends: start from the givens and work forward and start from the finish line and work backwards, and hope you meet somewhere in the middle. Working backwards is often more valuable than working forward when it comes to proofs, especially when they're of the less straightforward variety (i.e. there are too many ways that one could work forward, so you're unlikely to get to the finish line like that).

As far as books that discuss strategies like this, I recommend George Polya's short book "How to Solve It". There's also a great series of books called "The Art of Problem Solving", although those are more focused on proofs encountered in math tournaments.