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Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove?

Have you looked at some of Chaitin's work? The digits of his Omega number are an interesting case. His take on it is interesting: anything sufficiently complex is essentially random.

But on a more mundane level, any finite theorem proving machine (and I'm assuming people are that) will be helpless in the face of a theorem whose smallest proof exceeds the capacity of the machine.

I'm enjoying being a spectator to this debate so much, I think I'll join in. My own view is that the ability to do science is hardwired in our brains, and as such, may possibly be the result of evolutionary pressures. If are to assume that the neo-Darwinist approach to cognitive psychology is true (as I do) then we do NOT possess a general all-purpose brain which is capable of deducing or understanding everything. Instead, we have evolved very specific mental structures (or mental algorithims, or what ever term you prefer) which have proved useful to our survival and sustenance, and which through some way, some combination of adaptative and exaptationist processes, have been useful beyond their original purpose. And yet, the limitations are still there; there are certain phenomena which are minds are unable to ever completely discern or explain.

If this seems like an unduly pessimistic expression of biological determinism and fatalism, then allow me to conclude on an upbeat note. The biological apparatus allowing us to science encompasses our perceptual faculties as well as our cognitive ones. We have been able to move beyond the innate limitations of our five senses through the instruments we have devised-everything from telescopes to microscopes, thermometers to chromatographs, radio telescopes to bubble chambers. We may be able to similarly go beyond the biological limitations of our masses of gray mush by developing computers and other artifical intelligence systems capable of doing science, rather than simply serving as assistants. How this may be done-well, that's an answer which is beyond me.

That's an interesting website, thanks! I'll print out some of his papers and read them. I have heard the quote before that anything that is sufficiently complex can pass for random. Didn't realize he said it.

But it's still not clear to me that there are necessarily proofs out there that exceed the capacity of man's brain to understand, but I suppose that'll come out as I read some of his articles. It seems to me that he'll never be able to prove that a given theorem is beyond mankind's ability to prove. Of course, there are lots of unproven conjectures bouncing around out there, but when it comes to many of them, we just don't know if it's because we're not and never will be smart enough to prove or disprove them (I sure hope that's not the case, though it does appear to be what Chaitin is saying), or if it's because math isn't sufficiently developed to allow us to do much with them yet (another way of saying this is that the right genius hasn't come along yet and showed us the correct way to approach the problem), or if it's because it's one of Godel's undecidables.

Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove?

Consider the four-color theorem. The original proof requires the use of a computer to check billions of cases; here's a link to another proof, where the author says

"The first proof needs a computer. The second can be checked by hand in a few months, or, using a computer, it can be verified in about 20 minutes."

As a general rule, given any formal system, and given any positive N, there will be theorems in it whose proof requires at least N steps. That's any N - eg 10 to the 10 to the 10 ... to the 10.