But if a person accepts the Church Turing Thesis, then the human brain is not doing any computation that can't be modeled by a Turing Machine. In fact, the brain is at least a Turing Machine. Gödel's limitations will apply to it.
If we accept the Church Turing Thesis and replace the brain with a Turing machine, I argue that the mind is a program that runs on that Turing machine. The program embodies a particular formal system sufficient to encode mathematical statements and leverage itself to prove statements. From this argument one can infer that there are some mathematical truths that some minds will not find. And that if each human mind represents a different formal system, it is in their best interest to work together.
Disclaimer - The above is me thinking out loud not something from a peer reviewed paper
Which is not necessarily a bad position to take. http://xkcd.com/505/
Gödel's Theorem has been used to argue that a computer can never be as smart as
a human being because the extent of its knowledge is limited by a fixed set of
axioms, whereas people can discover unexpected truths
Ever stopped to think that the very "obviousness" of it might just be a product of professional bias as IT persons?
Trivially you can, with perfect information and sufficient technology, modelling neuron by neuron, create an electronic brain. The catch is, you would only prove humanOS runs on silicon as well as it does on carbon.
Having a powerful enough computing device does not imply it can compute anything a different type of computing device can. We have the turing-completeness that indeed says this is valid for a subset of our current programming languages and hardware, but turing-completeness does not trivially apply to our brain.
As an analogy, a ruler is a drawing device, but you can't draw a circle with it as with a compass. You can somewhat simulate drawing a circle by taking it very slowly using a certain clever algorithm, but it will never be perfect or else require infinite or virtually infinite time/space to do so.
That's not the case. If one accepts the Church Turing Thesis, then those functions the brain performs on effectively calculable functions can be performed on a Turing Machine. The brain is at least a Turing Machine, but may be greatly more than one, and Gödel's limitations will only apply to that portion doing calculations. The brain may very well be doing a great deal more than simple computation.
Thus, when you write I argue that the mind is a program that runs on that Turing machine, you're leaping off into wild speculation.
If the universe is not calculable on a Turing machine then some physical processes including those going on in the brain are not computations but can only be expressed using superrecursive algorithms. If those processes in the brain were computations then the human brain would be a hypercomputer. I do not believe in the existence of that latter. This opens the possibility that even if the brain operates via non computable means, its behaviour could be fully captured by a Turing Machine. I also think the theory that the universe has non computable things going on and the brain harnesses them in a non algorithmic way is more complex than the theory that the universe is merely Turing equivalent and so is the human brain.
My basis for this belief is the unrelated fact that there are some strict limitations in reality. Finite Speed Limit, 2nd Law, Maximum Force, Maximum Information per square meter, Quantum Indeterminacy; Compuational Indertermincancy of various facets: Diophantine, Church, Godel, Turing, Chaitin. Also the prudent belief that P <> NP and more importantly, lack of any evidence of Nature doing P in NP. Also: No Free Lunch in Search and its counter (okay no free lunch but the universe has structure exploitable by turing machines - see M Hutter). To me, saying the universe is just a turing machine fits this pattern.
Other patterns are the various links which occur in: physics, topology, logic and computation; the unifying power of category theory (e.g colgebras/algebras:objects---analysis as tagged unions---algebra), the link between physical and information entropy, the possibility of a Holographic Principle, the possibility of a discrete theory of quantum gravity, the relationship between a complex probability theory and Quantum Mechanics and the informational nature of QM. To me all these are very suggestive of a simple underlying nature which is informational and that digital physics may not be correct but it is in the right direction.
In both cases the problem is insufficiently respecting the rigorous boundaries on what the theorems actually say and apply to. Neither theorem has anything to say about consciousness and the human mind, unless a lot of currently unproven preconditions, some of which seem unlikely, get proven first.
I don't know, I'm just thinking out loud....
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.
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[1]: http://arxiv.org/abs/quant-ph/9907009
[2]: http://www.nature.com/nature/journal/v446/n7137/abs/nature05...
Extra reading: see [wikipedia](http://en.wikipedia.org/wiki/Orch-OR)
Not to dwell on arguments about whether or not time and space actually exist independently of human experience, what Kant was getting at is that the way in which humans experience the world limits our ability to draw conclusions to a particular subset of all truths.
If Godel's theorem is true, then from a Kantian perspective, mathematics no longer enjoys a uniquely privileged place in regards to human rationality. That's pretty important philosophically - at least to some people.
Positivists may take a different view.
One implication of GIT is that there are true sentences that cannot be proved.
I'm sorry, but I can't manage to see how "It has absolutely nothing to do with the limits of rational thought."
It saddens me that I am the first one that had to point this out on this comment that is over 15 hours old. On HN. :-(
I don't think this is true, at least not in the way you put it. I for one surely cannot work with the True Arithmetic system, because I cannot distinguish axioms from non-axioms. Maybe you could expand it a bit?
I would like to know the flaw in my reasoning.
[1] - http://en.wikipedia.org/wiki/Orch-OR#The_Penrose.E2.80.93Luc...