So the followin…
So the following disjunctive conclusion is inevitable: Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified . . . (Gödel 1995: 310)
Minds, Machines and Gödel, First published in Philosophy, XXXVI, 1961, pp.
The Lucas-Penrose Argument about Gödel’s Theorem, Internet Encyclopedia of Philosophy
To me it’s clear from the above quote that Gödel understands incompleteness as revealing that mathematics is not completable in the sense that it is not enough to contain the reality of human consciousness. I disagree with those who use incompleteness to suggest the opposite position, that incompleteness demonstrates the incompleteness of the powers of human approximation to contain the grandeur of computable truth. Certainly human understanding is limited, but that our understanding of the limitations of arithmetic mechanism is part of what falls outside of that limit.
Proving that we cannot prove ourselves consistent assumes, erroneously, that doubt is not also a form of belief which depends on an expectation of consistency. The mistake that is often made, in the Western mind’s eye, is that since belief in belief is the ultimate bad, then belief in disbelief must be the ultimate good. This bit of Manichean simplicity is exacerbated when the skeptic no longer sees their own skepticism as a form of belief, and takes it for granted that absolute doubt is possible, reasonable, and independent of unscientific bias.
Even the term ‘belief’ is a second order logic which presumes a first order doubt beneath any given feeling, thought, understanding, intuition, etc. We can see that we should question our own authority, but we forget that authority includes the very authority to question itself, and that such an inescapable authority can only be more primitive than either fact or fiction. Before fact can be wrestled from fiction, or fiction can be confabulated from fact, there must be a capacity to discern one from the other, and that capacity cannot be fiction. Descartes, in my view, didn’t go far enough in saying “Je pense donc je suis”, because it doesn’t specify whether I exist in thought, whether thought exists in me, or whether, as I suggest, thought and I are distinctions of sense which are within the primordial pansensitivity that underlies both uni- and -verse.
Instead of seeing the limits of our human perspective as evidence that all privacy is solipsistic and isolated, I suggest that our perspective is imperfect only to the extent that it is human. When we compare human perceptions to the low level common behaviors of measured objects, then there is a lot that we can learn from physics which we could not learn from human introspection alone.
The fallacy is to conflate our human ignorance with the superiority of measurement to sensation and to overlook that the ontology of measurement supervenes on some form of sensation. Once we compare (absolute experiential) apples to (absolute measurable) apples, we find that the latter cannot be more complete than the former. Physics and math are more complete than human experience, but as they are only experiences in which other experiences are reduced and measured to a generic abstraction, they are less complete than experience itself. No map of France actually leads to Paris, no matter how precise the directions are. A map of France can only contain a map of Paris, and a map of Paris can’t be Paris itself, because it is only a pattern built from generic measurements which do not know anything about Paris itself.