Gödel’s proof and the role of human belief
Just being yet another writer making far-reaching conclusions about Gödel’s theorem.
Ernest Nagel & James R. Newman write in Gödel’s Proof book conclusion:
[Godel’s proof] does not mean, as a recent writer claims, that there are “ineluctable limits to human reason”. It does mean that the resources of the human intellect have not been, and cannot be, fully formalized, and that new principles of demonstration forever await invention and discovery.
What the authors mean by the intellect being fully formalized is that our intellect would be described as a finite set of axioms and transformation rules allowing full capacity of our reason. The set of axioms being finite is crucial here. So I’m reading in this that our beliefs are the power of our reason – results that we don’t know for sure, but we believe in them. If we can’t prove Goldbach’s conjecture then we can simply believe in it and use it in our reasoning, like a new axiom. Still, Gödel showed that no finite set of beliefs (axioms) is enough to be able to prove everything in the system as “simple” as arithmetic. We just have to pick our beliefs and see where they take us.
Then we can pick beliefs and still maintain consistency. But, of course, the human mind neither is limited to consistent systems. Intellect can perfectly reason in inconsistent ways. We can juggle contradicting believes and still achieve results. This is seemingly so contrary to the premise of mathematics which tries to find and prove absolute truths.