Reducing P vs. NP to SAT
The way to view testing a math statement (e.g., a resolution of P vs NP) as a question of the form "is formula .. satisfiable" is the following:

Fix some axiom system.
Given a string of length n, whether the string is a proof for the math statement in the axiom system, is something one can define in a straightforward manner:
the string should consist of propositions. Each proposition should either be an axiom, or should follow from the previous propositions by one of the inference rules.

It's not a problem to define a Boolean formula that verifies all this.
All you should know is the length n of the proof!