instruction book that we wrote to describe physics?
There's the thing that you don't understand. We didn't create mathematics to describe physics,
I don't know quite who you mean by we, but Newton (along with many, many others) actually were trying to describe physics when they created mathematics. As stated above, it's not too much of a surprise that they eventually got it right.
If it's possible to enumerate all valid proofs I propose the following proving algorithm: Run through all valid proofs; once you get to a proof whose conclusion is the theorem you want to prove, return that proof.
[if you don't know whether your theorem is true or not, run the above algorithm on its negation as well].
What's wrong with my algorithm?
The problem is that some propositions P have the following two properties:
1: P has no proof
2: (not P) has no proof.
So your algorithm searches forever and you don't know if it just hasn't found anything yet, or if there is nothing to be found.
No amount of careful planning will ever replace dumb luck.