> The main point is the brain, as far as we know, is reducible to the known laws of physics, which, as far as we know, are entirely computable. Therefore, if human beings exhibit non computable phenomena, such as functioning as halting oracles, then that is a good reason to believe the mind does not reduce to the brain.

This is where I disagree. It would, to me, seem to be a good reason to believe we've missed something about the way the body (not specifically the brain) works.

> And we could call this 'mind' a new form of matter or something, so that people don't get alarmed by non-physical phenomena.

This just sounds like you are eager to label any new phenomena "non-physical". There is no reason to pull the extraphysical gun just because you are stumbling upon new physics.