Nice article. I'll add one more argument. Most people think the universality of Turing machines (the possibility of emulating one Turing machine with another) means that the universe must be a computer because if we can build computers then the universe itself must be some kind of computer by computational universality. But the problem with this is that nothing says that a non-computable universe is not allowed to have Turing machines nested within it.
A somewhat tortured analogy is inscribing a square inside a circle and then claiming that the circle is actually a square because squares can be inscribed within other squares. The folks that believe in a computational universe don't have a convincing argument for why they assume the universe is a computer other than assuming it as an axiom, which is fine but is not much of an argument for why the universe must be a computer.
Similar argument is true for the brain. Humans can carry out computational actions by emulating Turing machines but that is not a good reason to believe that the brain is also a computer because the same argument applies. Just because a Turing machine can be "nested" within biological brains does not mean that brains are themselves Turing machines.
I don't think one needs infinitly many states to store the state of the universe. Why? Quantum mechanics only need Hilbert spaces which are representable with a finite number of bits.
A somewhat tortured analogy is inscribing a square inside a circle and then claiming that the circle is actually a square because squares can be inscribed within other squares. The folks that believe in a computational universe don't have a convincing argument for why they assume the universe is a computer other than assuming it as an axiom, which is fine but is not much of an argument for why the universe must be a computer.
Similar argument is true for the brain. Humans can carry out computational actions by emulating Turing machines but that is not a good reason to believe that the brain is also a computer because the same argument applies. Just because a Turing machine can be "nested" within biological brains does not mean that brains are themselves Turing machines.