I haven't read Hostadter or the other's arguments, but this comment got me thinking. If the incompleteness theorem is about the limitations of an axiomatic system, why wouldn't it apply to rational thought. If (and it's by no mean certain) rational thought and the human brain have certain rules built into it to process, interpret, and act on information from the outside world, then couldn't these hardwired rules be considered axioms? These axioms are clearly powerful enough to express the integers and so one would conclude that there are certain propositions that are true but cannot be proven by the human brain. Of course, the brain and it's wiring is still a matter of research, and the fact that the brain is a dynamic system with neural connections in constant flux means that the system is not static nor are these "axioms".
Penrose's The Emperor's New Mind addresses this idea. Basically, Penrose assumes that:
1. Gödel's Incompleteness Theorems are equivalent to the halting problem (provable)
2. If the human mind is deterministic, it can be modeled with a deterministic algorithm (provable).
3. The human mind seems to be capable of proving arbitrary things about all algorithms (debatable; I'm extremely skeptical).
4. Therefore, the human mind is not bound by the halting theorem, and therefore the human mind is not deterministic.
He then draws several possible speculations. 1, the human mind is driven by quantum mechanics. He explains this more in Shadow of the Mind, the "sequel" to this book, and goes on to postulate specifically that "microtubules" allow quantum mechanics to have far-reaching effects that are indistinguishable from consciousness. Other people, most notably and substantially Max Tegmark, disagree[1], arguing, "we find that the decoherence timescales (∼10^−13 to 10^−20 seconds) are typically much shorter than the relevant dynamical timescales (∼10^−3 to 10^−1 seconds),
both for regular neuron firing and for kink-like polarization
excitations in microtubules. This conclusion disagrees with
suggestions by Penrose and others that the brain acts as a
quantum computer, and that quantum coherence is related
to consciousness in a fundamental way." Since then, quantum effects, especially quantum teleportation, appear to be crucial at the molecular level in processes like photosynthesis[2], suggesting that, perhaps, quantum mechanics may play a role.
However, even if quantum mechanics do play a role in human consciousness, I don't see how trading a deterministic brain for a random one is an improvement. Personally, I don't think that quantum mechanics do provide a crucial level to the extent that our brains are somehow not bound by logical axioms; I still believe there's a fundamental "algorithm" that drives the biological brain, though it may be difficult to conceive of, and I tend to agree with Tegmark in ideas of consciousness. Still, I'm open to any evidence on either side of the table, but I think we're a few decades away from key discoveries about the way our brains work that will shed any serious light on the subject.
> Just because you can create a function to return a row of Pascal's triangle doesn't mean you're a great programmer but if you can't, it almost certainly means you aren't. It's a negative signal filter, nothing more, but an incredibly quick and useful one.
Or as we say in the medical field, this test has high "sensitivity" :)
I took the algorithms class at UCSD a few years ago from Sanjay Dasgupta, one of the authors of that book. At the time the book wasn't finished but we used a draft as the lecture notes. One of the best classes I took. I use that book to this day as a reference for some algorithms.
I don't know, I'm just thinking out loud....