>One way of describing the Incompleteness Theorems (1931) of the Austrian logician Kurt Gödel is to say that he proved, in the form of a mathematical theorem, that the possibility of a fully automated mathematics can never be realized

What Gödel apparently missed (although it's hard to tell without going over his writings carefully), is that humans, by the same criterion, can't "solve" mathematical problems. This is essentially a re-statement of the halting problem (undecidability).

You can't expect to go to the best mathematician in the world (o.c. that's not a thing, but say for example Terry Tao), give him an arbitrary problem and expect the problem to be solved within a fixed time. Or even some arbitrary but finite time (even if he had infinite scratch paper). If he had an obligation to solve the problem (within an axiomatic basis), there could be problems that go unsolved. Any mathematician knows this -- there's no guarantee the Riemann hypothesis will be resolved -- it could be within a year, within a century, or never.

It's very attractive to think we have some kind of higher powers (that transcend ordinary matter, or computers, or w.e.), but we do not and mysticism is not necessary to explain our abilities.