Just to make things clear: The Church-Turing hypothesis has nothing to
do with some dogma of a hypothetical Church of Turing. There is
nothing religious about it. On the contrary, it is just a hypothesis
summarizing the best of our knowledge. There is no metaphysical
Implication. The question whether humans could do better, that they could
achieve more than machines, is a metaphysical question as we have
strictly no handle on it, no hint whatsoever of what could
differentiate a human from a machine. So this question should be
migrated to metaphysics.stackexchange.com.
But let us assume that the human brain can solve the halting problem
for Turing Machine. Then the computational model of Turing Machines
becomes much less important, and the Church-Turing Hypothesis becomes
much less relevant, as we have a more powerful model called the Human
Model (to avoid the word machine). Of course this (arbitrarily
long-lived) human model comes with its own hypothesis on
computability.
But then, while the halting problem for Turing Machines is no longer
critical, we now have to deal with the Human Model Halting problem.
And diagonalization will show that the Human Model Halting problem is
not decidable by a Human. Then what?
Now, you might object that diagonalization would not be applicable.
That would mean, I guess, that associating some form of Gödel
numbering with computing devices, proofs, or whatever we describe with
notation would no longer be possible, though it is currently the basis
of all science. In other words, we would have to deal with entities,
concepts that have no written representation, that cannot have a
written representation, or to say it more generally concepts without a
syntactic representation, whether written, oral or otherwise.
Of course, this would be in opposition with the teaching of John
whose very first sentence is: "In the beginning was the Word, and the
Word was with God, and the Word was God." Negating the fundamental
importance of syntax, of the word, is thus a very anti-christian statement.
I am of course not taking a stand on this, but since my first take on
this question is that it is a metaphysical one, and since the question
is not on hold, it seems natural to consider all consequences,
including the metaphysical consequences.
