[vidarh]

There's no "napkin philosophy" here.

What you call "systematizability" is irrelevant. Only whether or not a function can be computed matter, and if a function can be computed by a brain restricted to known physics, there is no reasonable theory under which it can compute anything which can't be computed by any universal turing machine. Given the absence of even any real hypothesis for any category of functions which can't, it'd be an absolutely extraordinary claim to suggest such a function exist.

Where you'd get a Nobel in physics if you could show unknown physics in the brain, you'd secure a Nobel in mathematics if you could come up with a class of functions the brain can compute which can't be computed by a computer, because it'd fundamentally break basic assumptions in any number of subfields of both maths and computer science.

What you describe as "subverting itself" does not require anything as drastic. It merely requires thought processes unobservable by the person in question, and processes to complex to casually understand and/or randomness. Given a large proportion of people don't even have an inner voice they can "listen to" in order to observe their thought processes other than via outputs, and we know e.g. from split-brain experiments that the conscious part of human reasoning often outright invents knowledge and observation of thought processes, we have no reliable way of analysing the extent to which thinking is "systematic", and plenty of evidence to suggest we can not trust any such notion.