One other subtle requirement of Godel’s Proof is a special use of numbers in which unique numbers are used to encode mathematical formulas and even other numbers. In this manner, numbers are used to describe themselves, a form of self reference. At one level the numbers are numbers, yet at the meta-level the numbers provide information about the truth or falsity of equations.
This use of self reference may in some ways have equivalence to our own mental representations making reference to our selves or the ability to observe mental states. Theoretically the electrical and chemical processes of the brain could be understood in a mechanistic way. Yet, the actual experience of being seems to elude the reductionistic approach.
The surprise of a new insight, perspective or understanding which was true but unknown may, in fact, be a rough equivalent of the true but unprovable Godel statement. These insights often come unbidden, or through self-reflection, or depth psychotherapy, but the surprise of the AHA moment creates a new whole of reality for the individual who experiences it. An interior depth of new views of a data set that is largely unchanged in detail, the meta-view of the self.
There is one key distinction to be made in this particular discussion regarding the concept of an infinite number of truths not provable within any formal system. Many post-modernists have taken Godel’s ideas to promote the primary tenet of post-modernism which is that all narratives [truths] are equally true. The corollary to their perspective is that various truths cannot be compared, especially if they arise from different cultures or worldviews.
From that disastrous series of errors, the post-modernist arrives at political correctness and other forms of idea restriction. The reason for this degeneration in communication is because truth is no longer the value that determines what can be argued since all truths are equal. The means the post-modernist then justifies is the emotional effect on the reader. Anything that potentially offends is forbidden under this schema.
As a logician, Godel would, I believe, have been horrified by this misuse of his proof. Godel would have easily known that there are actually infinitely more false statements than there are true statements in the universe of ideas. His proof does not proclaim false statements to be true, nor would he have ever endorsed the abandonment of logic itself. His proof only stated that there exist statements which are true, but not provable; he does not claim or endorse the concept that all statements are equally true.
One of Godel’s biographers, Dr. Wang, a rather conservative philosopher himself, concluded that the consequences of Godel’s Incompleteness Theorem for Mathematics included at least one of the following, if not all:
1. Mathematics is inexhaustible.
2. Any consistent formal theory of mathematics must contain undecidable propositions.
3. No theorem-proving computer (or program) can prove all and only the true propositions of mathematics.
4. No formal system of mathematics can be both consistent and complete.
5. Mathematics is mechanically (or algorithmically) inexhaustible (or incompletable)
Certainly if mathematics, as the foundation of science, is without limit, that at least suggests that other aspects of reality are also without limit, inexhaustible, and contain “undecidable propositions.” There is no reason that this limitlessness should be necessarily limited to mathematics. Godel actually believed that he had demonstrated the truth of Platonism, but neglected to publish that further proof. This proof certainly does imply that “truths” are discovered from a larger field of reality rather than merely created as an arbitrary convenience.
Godel certainly believed that although brain states might be mathematically determined as measured by such things as electo-encephalograms or brain imaging techniques, nevertheless, neither of those techniques nor any other mathematically based technique could [even in theory] predict or determine the richness of consciousness. He was certainly accurate regarding the limitations of the abilities of digital systems like computers to emulate consciousness, and was a consultant to the Artificial Intelligence community until his death.