Is it possible that there exists a category of proofs we have not yet discovered?

That’s like asking if our list of musical genres is exhaustive, or if there could be music beyond jazz, funk, pop, soul, rock, …

What a proof boils down to is a process that takes you from known facts to a new conclusion based on reasoning, or the rules of logic. A valid proof could in theory be reduced to a step-wise list that starts with some of the axioms of your mathematical system, and deduces other formal statements by applying formal inference rules.

But that’s like saying that you could reduce a song to a time-series of frequencies. Sure, that would capture the song formally, but that’s not how a musician works with music. When a mathematician tries to establish the truth or falsehood of a statement, he can employ one or several heuristics or “strategies” that mathematicians have developed shorthand names for. Where a musician might tell another that he might be able to establish a desired musical effect through a plagal cadence, or through syncopation, or through any other shorthand a musician uses for a general musical strategy, a mathematician might advise someone to try establishing a fact by induction, or by contradiction, or by infinite descent, or by construction, or by establishing an invariant, or by…