>Is the claim by Penrose not that humans can possibly intuit theorems which are true but unprovable (Gödel's first incompleteness theorem) e.g. the Riemann Hypothesis, and thus how could an algorithmic process ever achieve such intuition?
A "true but unprovable" Goedel statement is true in standard models of arithmetic but untrue in certain nonstandard models. The "incompleteness" is syntactic, not semantic. The real and complex numbers, AFAIK, only have one model, up to isomorphism.
And sometimes statements are difficult to prove because they're actually independent of the foundational system. Or because a counterexample exists somewhere.
"This problem is unresolved, therefore it's a Goedel Statement within our current foundational system" is extraordinarily unlikely. For one thing, that would imply that we could figure out the axiom we're missing and pass to the stronger system capable of resolving the conjecture straightaway, or that starting from some stronger foundation like homotopy type theory would resolve the conjecture right-away. Most unresolved conjectures are not unresolved for lack of proof-theoretic strength in our foundations.
