I agree with your sentiment. Gödel's work applies only to formal axiomatic systems of a specific strength. There are formal systems that are so weak they can prove their own truth and every statement is decidable (every true statement can be proved as true). And if your theory makes no claim to an axiomatic basis then Gödel really does not have jurisdiction.
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
It's a fallacy to think of a human brain as a distinct entity. If you are viewing the human brain as "just" a bunch of matter interacting in interesting ways to produce what we call consciousness (as opposed to viewing consciousness as something separate from the physical processes of the brain), then there is nothing really distinguishing your brain from the rest of you, and indeed from the rest of the universe. (Practically speaking, a good bit of decision-making happens in neurons outside of your brain.) So if you want to argue that the mind is equivalent to (or can be simulated by) a Turing machine, then you really want to argue that the whole universe could be simulated on a Turing machine.
I have the vague hope that this insight about the nature of the brain being a type of computing device might be kind of obvious to us as IT and information science people. It is, however, sadly not obvious to mathematicians and physicists as the article itself quotes:
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
...which to me is an utterly surprising non-sequitur.
Well, that paragraph you quoted is indeed non-sequitur, clearly you can program computers to discover unexpected truths using their 'axioms', code their 'axioms' to be as flexible and fragile as ours and so on, but the assertion that the nature of the brain is a computing device in the sense we "information science people" refer to one, is far from obvious. Obviously the brain computes stuff, but we tend to be rather specific and we think of computers when we say computing device, which might or might not be able to simulate human intelligence, hence the strong vs weak AI debate.
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.
I have the vague hope that this insight about the nature of the brain being a type of computing device might be kind of obvious to us as IT and information science people.
Ever stopped to think that the very "obviousness" of it might just be a product of professional bias as IT persons?
> 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.
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.
Excellent point and I suspected someone would call me out on this. Notice that I also said at least a Turing Machine. As I respond to another comment in this thread, implicit in my statement is the belief that the universe is calculable on a Turing Machine. I do not believe this is wild speculation. These ideas are not simple for me at least so I will be careful:
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.
The false conclusions philosophers, computer scientists and many others draw due to misunderstanding, or misapplying, the Church-Turing thesis are every bit as problematic as those false conclusions that result from misunderstanding or misapplying the incompleteness theorems. See http://plato.stanford.edu/entries/church-turing/
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.
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