@Voidmatrix
Indeed, I am still on a 'level'. But the higher the level I am on, the more levels I still have to go through to come back to 'level zero', thus the more 'work' I still have to do to discard them. Thank you for noticing.
I don't think there is anything in what I've written to be satisfied about for myself. On the contrary, my writings are generally borne of frustrations (which indicates some 'level' I guess). I do not claim to have any truths here. All I'm trying to do is to demonstrate falsehoods.
I can understand that my stuff comes across as arrogant. I could fluff it up a bit, but then I would lose some authenticity in the process. Better to be honest and arrogant than to be insincere and humble in my opinion. My arrogance does not lead me to believe that I'm right though. I love be wrong, this is how I learn. So please, I invite you, prove me wrong. In any case, my apologies if I have offended you here, this was not my intention, but perhaps it was essential to descend to the 'next level'
Unfortunately, my frustration remains. In the following, I will double down on my remark of irrelevance. So probably more arrogance coming, but it might give you some ammo to shoot me (or yourself) down.
Here we go:
Hofstadter himself says in his book 'Gödel, Escher, Bach' on p.25 (in my edition):
"His [Gödel's] idea was to use mathematical reasoning in exploring mathematical reasoning itself."
Also, he refers to Gödel's paper (1931) on the same page, where Gödel says:
"All consistent axiomatic formulations of number theory include undecidable propositions."
As Hofstadter himself points out, Gödel's work is essentially:
"the translation of an ancient paradox [Epimenides paradox] in philosophy into mathematical terms."
Surely, many have used this jewel of an idea since its inception to expand upon morelaborate philosophical and mystical ideas and concepts. But none of them tend to agree with another on several points to the degree that the pure underpinnings of Gödel's achievement largely become obfuscated. And if you do not entirely understand the many fault lines that distinguishes these interpretations (which I do not claim I do), it becomes really difficult to know what is what. Also, although Gödel has actually 'proven' his point within a specific realm of mathematical reasoning, these further elaborations do not necessarily have this quality.
The confusion of this already becomes apparent in the orginal Epimenides paradox from which Gödel's theorems are derived. Let's see. In Hofstadter's book, it is phrased like this:
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."
Hence, the self-referential paradox arises when you ask whether Epimenides speaks the truth.
But there is a rather obvious mistake involved here that many philosophers nevertheless have overlooked:
"The mistake made by Thomas Fowler (and many other people) above is to think that the negation of "all Cretans are liars" is "all Cretans are honest" (a paradox) when in fact the negation is "there exists a Cretan who is honest", or "not all Cretans are liars".
source"
Quite simply here, the paradox dissolves. This goes to show that whenever we leave the realm of precise mathematical reasoning, it is very easy to apply the paradox incorrectly.
Consequently, concerning Gödel:
There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more
controversial
source
Gödel was aware of this himself and was very cautious in applying his own work to 'bigger' metaphysical questions, which is described for example in this bit in relation to mathematical Platonism:
"Gödel was nonetheless inclined to deny the possibility of absolutely unsolvable problems, and although he did believe in mathematical Platonism, his reasons for this conviction were different, and he did not maintain that the incompleteness theorems alone establish Platonism. Thus Gödel believed in the first disjunct, that the human mind infinitely surpasses the power of any finite machine. Still, this conclusion of Gödel follows, as Gödel himself clearly explains, only if one denies, as does Gödel, the possibility of humanly unsolvable problems.
It is not a necessary consequence of incompleteness theorems.
source
Moreover:
"Sometimes quite fantastic conclusions are drawn from Gödel’s theorems. It has been even suggested that Gödel’s theorems, if not exactly prove, at least give strong support for mysticism or the existence of God. These interpretations seem to assume one or more misunderstandings which have already been discussed above: it is either assumed that Gödel provided an absolutely unprovable sentence, or that Gödel’s theorems imply Platonism, or anti-mechanism, or both."
source
From the above,
I conclude that it is thus not because his theorems sows doubt upon well-established systems of mathematics, reason and logic, that it automatically implies that anything beyond these systems of reason suddenly CAN be critically analysed or validated based upon such doubt. Although doubt can open up new avenues for attempted reasoning, it does not by itself lead to new systems of critical reasoning.
Since, in their pure form, his theorems are not exactly applicable to what is discussed here, I am rather inclined to think that ANY elaboration or application of his theorems in terms of the supernatural or 'higher' beings, be it god, ra, angels, ghosts, etc. will lead us nowhere.
Furthermore, and this is essential, I think the very reason to invoke 'whatever entity' is exactly to prohibit such critical analysis! What truths can we derive (formal or informal) when a diety is invoked?
But, taking this information from the above about multiple interpretations (which I am not at all familiar with) into account, it might be of value to refer to some specific interpretation(s) you intend to apply, although I very much doubt it is even possible.
You've mentioned symbolic logic as an example. I've studied symbolic/formal logic a bit a long time ago, so forgive me if I'm a bit off here, but the way I remember it is that symbolic logic is based on the assumption that the validity of deductive reasoning concerns
form, not content. So in this sense, indeed, it would be applicable to any discussion of language, whether the content is mathematical, religious, political, or sex drugs and rock & roll.
But although the content is deemed irrelevant in this system, the objective remains to establish criteria for consistency and validity. In this regard, Wolfram for example defines symbolic logic as:
"The study of the meaning and relationships of statements used to represent
precise mathematical ideas."
source
It's been a while since I've read Hofstadter's Gödel, Escher, Bach (great book btw, good source). I've skimmed through my notes and reread some passages but I can't find anything that opposes my statement of irrelevance or help me to understand your point. If possible, can you refer to a certain passage or clarify how Gödel can be applied here in the eyes of Hofstadter?
I feel caution is required here so we do not end up in some line of questioning similar to the medieval scholars who asked themselves the following question (which I think you are familiar with): "How many angels can dance on the head of a pin?" Please, let us try to avoid to fall into a 'Strange Loop'. But otherwise, why not. En garde!!
