It seems to me that there are two ways of looking at the theorems, my way and the wrong way. The wrong way claims that we are enclosed in the theorems, so to speak, such that we cannot know truth (or truth is essentially reduced to logical entailment, so becomes tautologous).
Conversely, my way claims that in understanding the theorems we transcend them in a way no machine could ever do. It is accurate to say that if the our minds were computers, we could never know it. Rather, we could only know what we are programmed to know, so we could never step outside the program and assess its truth.
As Goldstein explains,
No matter how complicated a "thinking" machine we engineer..., this machine will run according to hard-wired rules that can be stated in a formal system, and when we ask this machine to tell us what the true propositions are it will be able to do so only by seeing which propositions follow according to the rules of the system.
Therefore, there will always be "a proposition that eludes its grasp of truth," such that "No matter how we strengthen the machine, by adding in the previously elusive propositions as axioms, there will be yet another proposition that will elude it... but not [elude] us" (emphasis mine).
In short, we can deeply understand Gödel's theorems and thereby transcend them, whereas no machine can step outside its programing in this way. Rather, a computer is always in the loop.
In an important sense, transcendence goes to precisely what humanness is and does; it is not only what distinguishes us from the rest of creation, but Life Itself does the same thing on another level, except in a non-self-conscious way.
In other words, as Life transcends matter, the Intellect (or soul) transcends Life. This is what our pal Robert Rosen was all about.
Hmm. I wonder what he says about the theorems and how they relate to Life Itself?
Ah, here we go, from one of the amazon reviewers:
This book is a powerful critique of the reductionist and/or simulation (modeling) approach to the mind/body problem and the "what is life" question. Rosen [argues] that mathematical models -- and more generally, scientific rigor -- which ban impredicative loops from scientific discource, would not allow us to build what he calls a "new science," which is needed to account for life and consciousness.
Another reviewer claims Rosen's work "is unquestionably of the level of importance of Einstein's Special/General Theory of Relativity, or Gödel's Incompleteness Theorems. This is a grand claim to make, but once you read Rosen's work, you will see for yourself."
I can't say I understand everything Rosen says, but this much I do understand. But I understand it because he's simply articulating a Truth I -- and we -- all know in our bones. It is the Highest Common Nonsense.
Here we go. I love this: "The celebrated Incompleteness Theorem of Gödel effectively demolished the formalist program," for he "showed that, no matter how one tries to formalize a particular part of mathematics..., syntactic truth in the formalization does not coincide with (is narrower than) the set of truths about numbers" (Rosen).
Bottom line: qualitative is not just poor or fuzzy quantitative, and semantics -- meaning -- can never be reduced to syntax, or to explicit rules of order (as if, say, the meaning of Shakespeare could be reduced to grammar).
Rosen puts it as succinctly as possible, that One cannot forget that Number Theory is about numbers. There is no escape from numbers via numbers, but this hardly means there is no escape! For "There is always a purely semantic residue that cannot be accommodated by that syntactical scheme."
Little semantic residues like, oh, Life and Mind. Mathematical formalization is simply too impoverished a language to map the richness of reality. To paraphrase Rosen, biology only appears "soft" to a physicist because it reveals qualities that cannot be accommodated by its syntax. Life is not simpler than physics, but infinitely more complex. Physics deals with special cases, since the vast majority of systems in the cosmos -- including the Cosmos Itself -- are more complex than anything physics can cope with.
Reality comes first, the model second. Right? Unless you're a socialist or climate scientist. Rosen notes that "a constructive universe, finitely generated, consisting of pure syntax, is too poor to do mathematics in." (One more reason why God isn't a mathematician but meta-mathematician.) We know there will never be a scientific "theory of everything," because no matter how sophisticated, there will nevertheless be an infinite gulf between it and reality. No amount of fine-tuning will get one around Gödel.
So, "Thanks to Gödel's theorem, the mind always has the last word." The human mind "can alway go one better than any formal, ossified, dead system can" (John Lucas, in Goldstein).
Of note, the theorem "suggests that our minds transcend machines," while also making it "impossible to prove that our minds transcend machines."
This of course requires a different sort of proof, one adequate to the object (Subject) in question. Although mathematics surely "flows from God," it cannot lead all the way back up. Rather, there is always the leap of faith between math and reality, man and God.
Some people say you can't prove the existence of God. So what. Nor is there any formal proof of free will. Nevertheless, it not only exists, but is proof of God. Don't kid yoursoph: the only way out is via God, otherwise you are indeed just echoing around your own model. You might as well be insane.
"Just as no proof of the consistency of a formal system can be accomplished within the system itself... no validation of our rationality -- of our very sanity -- can be accomplished using our rationality itself.
"How can a person, operating within a system of beliefs, including beliefs about beliefs, get outside that system to determine whether it is rational? If your entire system becomes infected with madness, including the very rules by which you reason, then how can you ever reason your way out of madness?"
You can't. Thank God and thank Gödel.
A timely observation that describes the conspiratorial, Russians-under-every-bed left:
Paranoia isn't the abandonment of reality. Rather, it is rationality run amuck, the inventive search for explanations turned relentless.... 'A paranoid person is irrationally rational..., characterized not by illogic, but a misguided logic, by logic run wild' (Goldstein).