I am not provable in S
That is a wonderful paradox to subjective mind. I’m not about to tell you I know anything about mathematics or logic, because I don’t. Besides me being a lazy bum, not-knowing is why I’m here and not in academia writing papers and articles. Apparently I do not have the patience required to learn the formalities of a dicipline. Nevertheless I can spot an error in reasoning when I see one. For reasons unknown I have gained access to my objective mind as a complement to my subjective mind, and objectivity detects errors all over. The debate on artificial intelligence vs. consciousness is a big one in that regard.
One of my absolute icons in the history of consciousness is Kurt Gödel. A truly brilliant mind that brought Einstein to Princeton, according to Einstein that is. To say that Gödels incompleteness theorem is flawed would be false. It is indeed true if restricted to formal systems only. The problem arises when Gödels work is used as an argument for human mind as something more than an objective Turing machine. A lot of subjective minds have done that and you can’t blame them. What can be expected from subjectivity but to defend its sense of self as something more than an object? Because of that compulsive quest for self evidence, there is hardly any objective science or philosophy out there. Believe me, to know truth I keep to my zen books which are littered with objectivity while being almost unintelligable to a subject.
Anyway, the reason incompleteness of formal systems do not apply to humans is that we are not objective and consistent in a formal manner. To accept this is a hard one for an ordinary mind, but perhaps even harder for the advanced thinker. After all, academic performance is in large part a consequence of an ability to handle abstract reasoning, and what could be less objective than the abstract? So, in a sound formal system the statements based on theorems are on the one hand consistent, on the other consistently abstract. Therefore, the system can be internally true without having any truth value in reference to objective reality. It is simply so that 1+1=2 is formally true even if no one has ever observed this relation in objective reality. This distinction between objective reality and symbolic reality is of course universal and true for all symbolism including E=mc2.
One tricky aspect of this is that symbols themselves are a manifest form of objective reality simply because they exist. This is the pitfall where spiritual camps get confused and start talking about illusion as not reality. That is to say that the actual map of the terrain does not exist which is complete nonsense. Especially if I’m hitting them over the head with it, prefably a massive road book just to prove my point.
Now, Gödel geniously proved that a map can be true in every concievable way, but it cannot prove itself as terrain. In fact, that is the only statement that cannot be true because “I am the terrain” if true means “I am not a map” and that means all other statements are false. That’s pretty bad for a map, right? That is why adding this particular statement to the theorem will also fail. Obviously because if we’re starting out with the premise “I am the terrain” we will (a) throw the darn thing away and go look for a real map, or (b) say the map is lying since we can clearly see that it is NOT the terrain. Instead we have an agreement with the map/system wherein we respond to the map as if being terrain. In that, we can navigate in objective reality knowing our guide is basically a hoax, but it lies in a consistent manner which makes it it very useful. So believe me or not, a system that is consistently false and subjective can be of great value in objective reality. Guided by lies we create wonderful things, nicht wahr?
Not having studied formal logic, I’m not able to say anything about how Gödel said it, but objectively, that should be it. The details are, from my perspective as a Turing-esque Response Machine, of less importance. I am way much simpler than all of the equations. The focus here is on the point of deception, that is; Why can I detect the truthfulness in a system declaring that it is per definition false?
That is because I myself am a paradox in stating: I am True. That is the subjective statement made by all believers in agency, intention, free will and causality. “I am”, we say, without being the least clear about the basic premises which would make such a statement true. Subjective self produces myriads of these abstract premises in order to stay true, but none will ever survive a single blow of an objective ten ton hammer of truth. I will in due time let the hammer fall on these attempts to produce a premise valid enough to save the self-statement, but for now it is suffice to say, they all fail miserably.
Descarte’s mistake was not knowing who it was that said “I think, therefore I am”. The objective statements are:
I think “I think, therefore I am”, therefore I think “I am”, in explaining subjectivity.
I think, therefore I am thinking, in explaining objectivity.
So human mind will detect the truth about the lie because human mind does not know itself correctly. The reason for that will be revealed in a follow up post, because right now I’m too tired to elaborate. When I’m paid to read all the references properly and given time to write a decent paper, this inconvenient truth could perhaps have some impact on advancing our common knowledge base, but I don’t see that day coming. I write this just to remember myself of who I am, besides my subjective self.
Final note, the Gödel sentence confuses formal/objective systems with human/subjective systems, therefore it tells us nothing about artificial intelligence. Grant me one year, a room with a view and three geeks and I will give you Subjective Intelligence. To know yourself as a fool makes that a pretty straight forward task. Believing yourself to be something more makes it impossible.
Just sayin’, and now I have once more offended any potential respondent enough to be totally ignored. That is what keeps me awake at night, not being able to adjust the obvious truth to the prevailing flawes, not being able to lie in a truthful way.
Maybe a golfer would get it if I said that a flawed swing that is 100% consistent and repeatable would make you a fortune. All you have to do is to adjust your stance a wee bit and leave everything else as it is. Then you’d hit the sweet spot with every shot.
It’s all physics and mass that matters.
Cheers to Kurt, I’m pretty sure he knew way more than he was able to carry on his own. I wish he’d known himself better…
Ooops, before logging out, I happen upon this one: Before Gödel’s work it had been widely conjectured that any precisely formulated mathematical yes or no question can be decided by the mechanical rules of logical inference on the basis of a few mathematical axioms. In 1931, Gödel showed that, no matter what axioms are chosen, there exist number-theoretical yes or no questions not decidable from the axioms. Combining this proof with A.M. Turing’s theory of computing machines, one arrives at the following conclusion: either there are infinitely many number-theoretical questions which human reason is unable to answer or human reason contains an element which, in its action, is totally different from any finite combinatorial mechanism and its parts. Gödel hopes it will be possible to prove that the second alternative holds.
–Paul Benacerraf, James S. McDonnell Distinguished University Professor of Philosophy, Princeton University
If this is not already done I’d be happy to offer such a proof, even if that element of reason seems unresonable. I believe Gödel would approve to fantasy as being the prime mover of mind.