I don't think this argument holds water for a number of reasons:
1. It is an unknown whether a finite volume of space can fundamentally be described by a finite number of states. You can extrapolate to this situation from your favourite theory, but this is not evidence that reality actually works like that. Physics has a long way to go to understand space-time completely.
2. Even assuming that is true, brains are not isolated systems. They are entangled with their environment. Why are you so sure that human cognition can be neatly separated into a finite box like this? The reality is almost certainly more complicated.
3. Lastly, you cannot measure a system like the brain to fundamental levels of detail without destroying it. You literally cannot clone a brain state if you take modern physics seriously, so this whole thing is a non-starter anyway.
You are right that we don't have a theory of everything. However, we can go pretty far with what we have and some clever reasoning. Eg when you have two different gases, like hydrogen and helium, in separate containers and mix them, you can build a relatively simple engine to extract work from that mixing.
That engine also works when your gasses are almost but not quite the same, eg when you have deuterium and hydrogen.
But it doesn't work, when you have the same gas, like hydrogen in both sides.
That gives a pretty strong hint that hydrogen atoms 'have no hair', ie they are all the same.
Fair point on 1, although I personally view that as extrapolation from an incomplete theory. But perhaps I have not read enough of the evidence for that bound.
I think you underestimate the complexity of systems with more than a handful of interacting particles. Yes, you can extrapolate with what we have but I think the answers you get are likely to be totally wrong outside of a very controlled regime. People simulating large scale interaction have to make a lot of simplifying assumptions, which need to be tuned to the problem at hand to be effective. Non-equilibrium statistical mechanics is poorly understood.
They make a lot of simplifying assumptions, because they don't just care about computability, but about finishing runs of their simulations before the heatdeath of the universe.
Our discussion was about the computability only, wasn't it?
1. It is an unknown whether a finite volume of space can fundamentally be described by a finite number of states. You can extrapolate to this situation from your favourite theory, but this is not evidence that reality actually works like that. Physics has a long way to go to understand space-time completely.
2. Even assuming that is true, brains are not isolated systems. They are entangled with their environment. Why are you so sure that human cognition can be neatly separated into a finite box like this? The reality is almost certainly more complicated.
3. Lastly, you cannot measure a system like the brain to fundamental levels of detail without destroying it. You literally cannot clone a brain state if you take modern physics seriously, so this whole thing is a non-starter anyway.