I'm sorry, I'm not sure what "unsolvability" means. What I'm saying above is that humans can identify undecidable statements, i.e. we can recognise them as undecidable. If we couldn't, Gödel, Church, and Turing would not have a proof. But we can. We just don't do it algorithmically, obviously- because there is no algorithm that can do that, and so no computer that can, either.
But that's the thing, humans can't do it either, except only in some very specific simple cases. We're not magical; if we had a good way of doing it, we could implement it as an algorithm, but we don't.
I'm confused by the discussion in your link. It starts out about decidability and it soon veers off into complexity, e.g. a discussion about "efficiently" (really, cheaply) solving NP-complete instances with heurstics etc.
In any case, I'm not claiming that humans can decide the truth or falsehood of undecidable statements, either in their special or general cases. I'm arguing that humans can identify that such a statement is undecidable. In other words, we can recognise them as undecidable, without having to decide their truth values.
For example, "this statement is false" is obviously undecidable and we don't have to come up with an algorithm to try and decide its truth value before we can say so. So it's an identification problem that we solve, not a decision problem. But a Turing machine can't do that, either: it has to basically execute the statement before it can decide it's undecidable. The only alternative is to rely on patter recognition, but that is not a general solution.
Another thing to note is that statements of the form "this sentence is false" are undecidable even given infinite resources (it'd be better to refer to Turing's Halting Problem examples here but I need a refresher on that). In the thread you link to, someone says that the problem in the original question (basically higher-order unification) can be decided in a finite number of steps. I think that's wrong but in any case there is no finite way in which "this sentence if false" can be shown to be true or false algorithmically.
I think you're arguing that we can't solve the identification problem in the general case. I think we can, because we can do what I describe above: we can solve, _non-algorithmically_ and with finite resources, problems that _algorithmically_ cannot be solved with infinite resources. Turing gives more examples, as noted. I don't know how it gets more general than that.
Sorry, but I really don't understand your claim. You say
> The only alternative is to rely on patter recognition, but that is not a general solution.
but then you say
> we can solve, _non-algorithmically_ and with finite resources, problems that _algorithmically_ cannot be solved with infinite resources
How do you propose that we humans solve these problem in a way that both isn't algorithmic and isn't reducible to pattern recognition? Because I'm pretty sure it's always one of these.
The first sentence you quote refers to machines, not humans, and machines must be somehow programmed with, or learn patterns from data, that's why I say that it's not a general solution - because it's restricted by the data available.
I don't know how humans do it. Whatever we do it's something that is not covered by our current understanding of computation and maybe even mathematics. I suspect that in order to answer your question we need a new science of computation and mathematics.
I think that may sound a bit kooky but it's a bit late and I'm a bit tired to explain so I guess you'll have to suspect I'm just a crank until I next find the energy to explain myself.
