[tjradcliffe]

Since the brain is not just a Turing machine, it would be astonishing if it was limited to the capabilities of a Turning machine. The brain is capable of emulating a Turning machine, but that is just one of its capabilities.

The things a Turing machine lacks are interupts and I/O. If you hook sensors up to a Turing machine and make the tape change depending on the state of those sensors, you don't have a Turing machine any more, in the sense that it is no longer bound by any of Turing's computability theorems.

This insistence that the limits of the most limited model of computing (Turing machines) must be applicable to any machine that computes anything--including things that are not describe by any of the formal mathematical proofs on the limits of computability because they violate the most basic assumptions upon which those proofs depend--is one of the most curious aspects this debate.

So not even computers are just Turing machines, because they too have I/O. Turing's theorems are useful and important when considering certain practical questions of computability within the limited circumstances of a computation whose inputs are entirely specified at the start and which can't be interrupted by new information coming from the outside, but they just don't apply to cases where there are cells whose values aren't known until reality provides them via some sensor mechanism.

As such, it would be a little weird if we (and computers) can't compute things a Turing machine can't, given we have capabilities that a Turing machine doesn't have. There are even examples of such things. One due to Church (IIRC) that shows how we can solve certain instances of the halting problem that a Turing machine can't.

[pjungwir]

Actually Turing's paper "On Computable Numbers" distinguishes between "automatic machines" (a-machines) and "choice machines" (c-machines), where the latter can pause to ask for input. It seems to me this accounts for your I/O. (I think this is also how you'd add a RNG to a Turing machine.) His paper pretty much only considers a-machines, so I'm curious what is written about c-machines elsewhere. It's unclear to me how much c-machines "change the story." For instance you can't escape Godel's Incompleteness Theorem by adding a finite number of axioms to fill in all the unprovable statements, because there are infinitely many of them.

[Phemist]

Unless interrupts and the Input portion of I/O are generated by an oracle (A machine of a computational class more powerful than UTMs) of some sort, a Turing Machine with I/O and interrupts is equivalent to a UTM. I remember the proof being trivial, ie some guys did it during my CS undergrad for a short (2-week) research introduction course, but I can't find a proper paper about it atm. Granted, if the brain was an oracle, this would be true, but you would have to presume that the brain was an oracle in order to prove that it is stronger than a UTM.

In regards to your last statement, the other way works as well. There are statements that can't be proved by any human brain, but can be proved by non-brain logical systems. For example: "The brain cannot consistently assert this statement". One can then create instances of types of problems that are equivalent (when put through some bijective transformation) to the beforementioned statement.