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It makes me personally think of a few passages from Rebecca Goldstein's book about Kurt Gödel ("Incompleteness - The Proof and Paradox of Kurt Gödel"). I guess my attitude about that statement depends very much upon "how much" truth we're shooting for, and if it matters to you that there might be an endpoint to the game, that we can't overreach, with a vast array of truths still "out there" forever beyond our reach. If you're going for broke so to speak, assuming that in principle it's possible for us to zoom in on each and every truth, then I do think that assumption must always be a matter of faith not proof, precisely because of Gödel's work. Here are the passages that statement made me think of. Mull over them a bit, and I think you might get the gist of why I feel the way I do:

From "Incompleteness - The Proof and Paradox of Kurt Gödel" (pg. 202):

Gödel himself was far more reserved about drawing conclusions concerning the nature of the human mind from his famous mathematical theorems. What is rigorously proved, he suggested in his conversations with Hao Wang as well as in the Gibbs lecture that he gave in Providence, Rhode Island, 26 February 1951 (which he never published), is not a categorical proposition as regards the mind. Rather what follows is a disjunction, an "either-or" sort of proposition. That is, he was admitting that nonmechanism doesn't follow, clean and simply, from his incompleteness theorem. There are possible outs for the mechanist.

According to Wang, Gödel believed that what had been rigorously proved, presumably on the basis of the incompleteness theorem is: "Either the human mind surpasses all machines (to be more precise it can decide more number theoretical questions than any machine) or else there exist number theoretical questions undecidable for the human mind."

What exactly did Gödel have in mind with this second disjunct. I "think" that what he is considering here is the possibility that we are indeed machines -- that is, that all of our thinking is mechanical, determined by hard-wired rules -- but that we are under the "delusion" that we have access to unformalizable mathematical truth. We could possibly be machines who suffer from delusions of mathematical grandeur. What follows from his theorem, he seems to be suggesting, is that just so long as we are not delusional as regards our grasp of mathematical truths, just so long as we do have the intuitions that we think we have, then we are not machines. If indeed we truly have the intuitions that we do, then it is impossible for us to formalize (or mechanize) all of our mathematical intuitions, which means that we truly are not machines. Of course there is no "proof" that we know all that we think we know, since all that we think we know can't be formalized; that after all, is incompleteness. This is why we can't rigorously prove that we're not machines. The incompleteness theorem, by showing the limits of formalization, both suggest that our minds transcend machines and makes it impossible to "prove" that our minds transcend machines. Again, an almost-paradox.