So, just what are we talking about?

[kandrathe]

Anyway, not meaning to poke at this dog food. I thought that rather than inadequately trying to explain something as complicated as why I think the way I do about Gödel, I would be better off to refer you to where I developed my opinion. I build robots, and I try to keep current with the frontiers of AI. I'm frustrated by the Minsky approach, and so I've turned to looking more at neurosurgery and brain model research which led me to Roger Penrose.

Roger Penrose claims that this (alleged) difference between "what can be mechanically proven" and "what can be seen to be true by humans" shows that human intelligence is not mechanical in nature. This view is not widely accepted, because as stated by Marvin Minsky, human intelligence is capable of error and of understanding statements which are in fact inconsistent or false. However, Marvin Minsky has reported that Kurt Gödel told him personally that he believed that human beings had an intuitive, not just computational, way of arriving at truth and that therefore his theorem did not limit what can be known to be true by humans.

So, I take it that even Gödel thought that there was a difference between mathematical "Truth" and the "truth" arrived at by human intuition.

Also, on Wikipedia I stumbled on this reference to something that had left a residue some 20 years ago when I had studied about all this. Presburger Arithmetic Which if I understand correctly is axiomatically complete and provable. Gödel's theorem requires the axiomatic system to be sufficiently strong, which Presburger Arithmetic is not. Anyway, it leads me to some doubts in general about the universality of Gödel's theorem.

And this tidbit is interesting on Peano Axioms;

In 1936, Gerhard Gentzen proved the consistency of Peano's axioms, using transfinite induction.

All mathematicians assume that Peano arithmetic is consistent, although this relies on intuition only. However, early forms of naive set theory also intuitively looked consistent, before the inconsistencies were discovered. This has been a source of confusion for a number of people, especially nonmathematicians.

The point is that we do have to rely on our intuition, and that it brings something new. Roger Penrose has argued that this intuition is what differentiates men from machines, but his arguments are dubious. The modern set theory often considers axioms postulating existence of large cardinals - none of them can be proved within set theory, nor is it possible to prove consistency of these axioms. But mathematicians generally do not exclude the possibility that some of these axiom systems are inconsistent. The intuition here is much less clear than in the case of natural numbers. Some people argue that even Peano arithmetic could be inconsistent - since intuition is not really a reliable source of truth. This argument can be extended and make us doubt even finite logic itself - these questions go back to Kant and his famous Critique of Pure Reason.

So, that's some of what has formed my opinion, based on my interpretations and shadowy recollections -- but, then again I may be on the side of the misled.