> It says nothing about the (information theoretic) entropy of the _dynamics_ of the brain, which is where the process of cognition actually occurs. In fact many (if not most) physical processes have infinite entropy.
I'm not sure how you reached this conclusion, assuming you're not just making a trivial observation that our formalisms employ the reals, which in itself entails nothing meaningful given the reals are merely a convenience, not a necessity.
If we accept that any bounded volume contains finite information I, and the laws of physics themselves have a finite description, then such laws are isomorphic to a state transition function I(t) -> I(t+1), which is readily simulable on a Turing machine. Of course, the devil's in the details of "finite" above.
> If the time required to reach another bit of precision grows faster than polynomial time, no Turing machine can hope to simulate that physical system in a practical amount of time.
I assume you're discussing the feasibility that cognition is simulable on standard computers here. I dispute your position that there's likely to be any quantum weirdness beyond ordinary chemistry, but even setting that aside, you're ascribing far too much power to the human brain.
Fact 1: natural selection would weed out any species with an organ that consumes more energy than any benefits it yields.
Fact 2: the growing evidence of many animals using tools, and our growing acceptance of a spectrum of consciousness among the animal kingdom, entails that human brains aren't too special, just special enough.
Fact 3: the detailed documentation of sensory defects, like optical and other sensory illusions indicates that whatever information processing algorithms we utilize, they are imperfect at best. This is exactly what we'd expect from evolutionary pressure, ie. we'd expect the most efficient information process that yields reproductive success, which is a flawed set of efficient heuristics that cover the majority of cases with the fewest resources needed, and not an energy-intensive, complicated, subtle but foolproof system.
Fact 4: many of the feats that were the exclusive domain of humans, like visual recognition and imagination, natural language translation, novel musical and written composition and more can be performed by computers. They're at roughly a 3 or 4 year old's level granted, but given we take 20 odd years to reach mastery levels in these disciplines, I'd wager that the computers will reach parity in that time frame too.
There's frankly little reason to believe the brain is special in any particular way, simply complex and layered, which will take some time to disentangle. It's like asking to describe the exact path of
> PS: To me all of the above suggests that whatever the universe is doing when it's doing physics, it certainly isn't computing in the Church-Turing sense.
Coincidentally, 't Hooft just released his book on the cellular automaton theory of quantum mechanics [1]. I certainly hope it will change your mind, because I think people these days far too readily dismiss determinism, and even the strong Church-Turing thesis, often for specious reasons.
[1] https://www.reddit.com/r/Physics/comments/6ntbvy/the_cellula...