Godel’s Contributions to Consciousness Part 4

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.

Godel’s Contributions to Consciousness Part 3

So what is this Incompleteness Theorem?

At the beginning of the twentieth century, mathematicians assumed that all of mathematics was a created form [Constructivism] simply utilized to express relations between things, whether or not those things were present in reality. On that basis, it was further believed that if all of the rules of this creative form could be fully expressed that all of mathematics could be known, and all future assertions in mathematics could be determined true or false based upon the formal system developed to do such. An amazing attempt at this process was completely by Russell and Whitehead in the three volume set, the Principia Mathematica. Formalism or Constructivism was assumed to be reasonable, true, and the future of mathematics, and to some degree logic. For the formal system to be functional, it would have to correctly identify all true statements accurately, and create no contradictions

Kurt Godel shocked the mathematical world be creating a short Proof, which demonstrated beyond doubt that all formal systems are necessarily incomplete. That in some manner a true statement can be introduced into the formal system, which the formal system could not identify as true. As it turns out, this Godel Phrase actually creates an infinite number of true statements that the formal system cannot identify as true, making any formal system necessarily incomplete, because, once the first Godel number/phrase is inserted, another [G’] is created, which continues without ceasing. Godel’s intense introversion led his announcement of the proof to be less than overwhelming, but other mathematicians of the day made the findings more public within the mathematics community.

Godel’s original proof made the concept of a formal system quite strict, such as the Principia Mathematica level of formalism. Perhaps had he stopped at that point, his work would be non-relevant to the questions of consciousness, but in a later version of his proof, he was able to so simplify the definition of a formal system, while retaining the result of the proof, that this otherwise esoteric bit of mathematics suddenly becomes substantial in a whole host of other intellectual inquiries. These true but unprovable statements, and their existence, have, in Nagel and Hoffstader’s words, forever separated the concepts of “true” and “provable.” That is an amazing paradigm shifter for any of us.

Consider: is it not the case that a creative idea which in every way seems fresh, intriguing and true after receiving the insight, would not have been necessarily identified as “true” by your mind prior to that creative insight?

Godel’s Contributions to Consciousness Part 2

Kurt Godel was born in 1906, and was such a tenacious child that his family referred to him as “Mr. Why” by the time he entered grade school. He made rather amazing contributions to the field of logic and mathematics over the course of his career. Einstein remarked that his greatest pleasures late in life were the daily conversations he shared with Godel, someone, it appears, he considered an intellectual equal. Godel provided Einstein with a mathematical solution to the field equations of general relatively, which he gave to Einstein at his 70th birthday. It was Einstein who helped him obtain a position at the Institute for Advanced Study.

 

Godel believed that his proofs confirmed Platonism, according his biographer, Wang, but he never published a formal proof of that assertion. He was known as an idiosyncratic person, and appeared to have starved himself to death in 1978 over a general paranoia of food. Irrespective of his personal oddities, his genius at logic has earned accolades that he was the greatest logician since Aristotle. The most relevant of his proofs for this discussion are the Incompleteness Theorems, which will be the subject of the next entries. These theorems, and indeed Godel and his work in general, were made part of public knowledge with Hoffstadter’s “Godel, Escher & Bach.”

 

Godel’s Contributions to Consciousness Part 1

Many of the cutting edge thinkers in consciousness studies refer back to Logician and Mathematician Kurt Godel. Examples include Douglas Hoffstadter, in his epic “Godel, Escher & Bach: an Eternal Golden Braid.” David Chalmers in his “The Conscious Mind.” Roger Penrose in his trilogy of works on mind brain interaction. These three alone account for some of the most intriguing concepts in advanced ideas related to consciousness.

Each of them note that based upon Godel’s Theorem [subject of an upcoming entry], it is not possible, even theoretically, for the mechanical predictable aspects of the electical/chemical brain to account for all the qualities associated with “mind.” Godel predicted the limitations of artificial intelligence in digital computing that have proved to be quite accurate, at least to date. The limitations he suggested have remained solid for the nearly fifty years since his death.

The three above authors all attempt to restore as reductionistic and physically based a theory of concsiousness as possible, given the constraints of Godel’s Theorem. Chalmers and Penrose actually wrote that the limitations provided by Godel’s Theorem could imply a more idealistic or mystical philosophy, but they specifically chose to limit themselves to a more reductionistic explanation. I would support a more radical approach, approximating that of Amit Goswami, a physicist who wrote the rather stunning “The Self Aware Universe.”

More to come in the next series of posts.