This is a silly objection, trivially because obviously no finite physical system - brain or computer or whatever - can be constructed with the storage equivalent of an infinitely long tape. But if you allow for the fact that humans can do things like write things down and share information with other humans and build computers to store information, our information processing capacity is not limited to the set of states we can hold inside the atoms inside our head.
But also, the claim lacks evidence: We’ve never seen a human being yet whose program didn’t eventually halt.
That doesn’t mean the hardware isn’t capable of running a program that never halts, just that we haven’t found such a program yet.
Indeed if you consider human mindware as a whole, given that when humans reproduce they create new copies of the mind running in new bits of hardware... maybe Human minds are infinitely recursive after all?
Technically Turing completeness requires infinite memory for that (or an infinite tape if we're talking about the original turing machine concept), which no Turing-complete machine has. In other words, the brain is as Turing-complete as any machine that we also consider to be so. We'll always be bounded by limited memory and limited time.
We do not know if human brain is indeed Turing-complete, or even if it is a Turing machine at all. Human Mind certainly is, but if brain is or not we do not know.
While I agree that comparing a human brain or mind to a Turing machine is not helpful, the objection you make here is less significant than it first appears.
There is a subtle difference between unbounded recursion, which a Turing machine is taken to be capable of, and the actual ability to achieve infinite recursion. In no application of a Turing machine, either as an actual physical device or as a hypothetical one in a logical argument, is it ever required to perform infinite recursion, which would just be one way of not halting.
For all practical and theoretical purposes, what matters is that the machine being considered does not exhaust its ability to recurse while performing the computations being considered. Consequently, the standard practice, of saying that computers and certain other devices are Turing-equivalent, with the usually-implicit caveat of being so up to the limit of their recursive ability, is both reasonable and useful.
> For all practical and theoretical purposes, what matters is that the machine being considered does not exhaust its ability to recurse while performing the computations being considered.
You're right, and thanks for the more strict definition.
Regardless, the 'recursion limit' of the human brain is really low. (Say, seven things at once or thereabout; not going to links proofs but it's a non-controversial statement.)
Certainly not enough to implement any sort of computing machine. Human brains are notoriously bad at arithmetic and state machines.
But also, the claim lacks evidence: We’ve never seen a human being yet whose program didn’t eventually halt.
That doesn’t mean the hardware isn’t capable of running a program that never halts, just that we haven’t found such a program yet.
Indeed if you consider human mindware as a whole, given that when humans reproduce they create new copies of the mind running in new bits of hardware... maybe Human minds are infinitely recursive after all?