I see nothing that leads me to believe that we're capable of grasping the complete truth in mathematics.We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths.
As I read it.
I was responding to this specific quote from Yourgrau (my underlines):
... The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers.
So, by this argument, we'd have to grasp the complete truth of mathematics (whatever that might mean) in order to conclude that "we, or our minds, are not machines or computers." I was calling into question the likelihood of our being able to grasp the complete truth of mathematics. And, if we can't, it no longer follows (from this argument) that "we, or our minds, are not machines or computers".
Good to hear from you, bud...
I agree with your analysis of Yourgrau's assertions. I disagree that he accurately stated Goedel's view.
Surely, ONE non-provable truth would be enough to distinguish oneself from a well-progranned IBM 360(remember those?).
Thanks for posting - I'd never run across this material otherwise.
Nevertheless, there are large parts of mathematics that are catagorical. Not only are all theorems provable, there is a proof scheme for these theorems. Euclidean (and consequently, all the non-Euclidean) geometry is one such. First order logic is another as is Pressburger arithemetic.