9. The Incompleteness Theorem of Kurt Gödel
- Children of the Primes -
Gödel's shocking discovery.In 1931 a grim and paranoid genius named Kurt Gödel - the Martin Luther of mathematics - discovered another reason why mathematics can not be the language in which the whole Truth may one day be expressed.
Probably as poorly understood and often misapplied as Luther, Kurt Gödel made himself odious by mathematically proving that a system founded on axioms - which are your basic assumptions upon which you build your understanding or theory or religion, etc - can never hope to find proof for the consistency of those initial axioms, and will therefore always be incomplete. This is called Gödel's Incompleteness Theorem and this theorem caused a Reformation in logic philosophy. Just like Luther proved that salvation comes through faith and not works, Gödel proved that the whole Truth can only be believed and not proven.
What a bummer for all those hopeful believers who believed that one day, somehow, either Math or their philosophy, or their religion would lead them out of the bondage of ignorance. No way, Jose...
Since Gödel we know that a logical system can not liberate. But that doesn't mean that rules are bad. Rules help create order and understanding. A society based on rules is far better off than a society not based on rules. And Math put a man on the moon. Math gave us the Internet. Math gave us this whole wild global culture. Yea, Math may run with the wind and the free range chickens as long as not the whole Truth is addressed. A system of logic can not cover Truth; Truth can not be expressed in logic. The Grand Unified Theory (GUT, or GUTH as a certain somebody demands) will not be written in Math. This is also the reason why you never hear anyone solemnly swear to tell the Truth, the whole Truth and nothing but the Truth, so help me Mathematics.
Still, she is beautiful. Do thy best old Math, despite thy wrong.
A logical system (scientific, philosophical, religious, legal)
There is, however, nothing wrong with being learned. Paul was learned. And so was Jesus. Solomon was a brilliant poet and philosopher, as well as an economical genius, way ahead of his time. And he wrote, "Trust in the Lord with all your heart and lean not on your own understanding (Prov 3:5)."
We've known for thousands of years but since Gödel we have a proof written in mankind's most common scientific language, fully in the tradition of Ezra, and making it less and less attractive to believe in anything else than God.
A clear and present example of the fallacy of logic systems is of course the number sequence itself. From a few simple axioms an infinite sequence is wrought that will never be water tight and new primes must inveterately be added.
The exact same pattern can be found in math and science. New ideas and new rules must continuously be added, and after these new rules have run their stretch, holes appear in the continuity of that which is known. Hence new rules must be added.
The western legal system dito. "To prove this let facts be submitted to a candid world". "We hold these Truths to be self evident." But as the world turned it became clear that even a document as profound as the Constitution could not cover all possible situations. Hence the Amendments and the Supreme Court which must add new explanations and interpretations to fill the holes in the continuity.
Bottom line: No logical system will ever be done forming and can never cover everything and can never represent Truth.
Hold that thought (9)
The most obvious Biblical application of this principle is of course the building of the Tower of Babel, in Genesis 11. The people that lived on the Plain of Shinar could not reach heaven by stacking brick upon brick. But the principle shows up all over Scriptures. In Luke 17:20 for instance where Jesus says, "The Kingdom of God does not come with your careful observation."
This has a very serious implication because it tells us something about the inherent qualities of Truth even before we understand what Truth actually is.
But as axioms lead to an unfixable chasm, the lack of such a chasm will not lead to something like axioms: unexplainable assumptions or initial and self-evident truths that would force the observer into one specific initial position. In Truth there is no specific initial position. There's only freedom. Where axioms will always miss their intention, Truth will always set free.
We have established that Truth is initially singular. And we've also established that a system that leans on the stacking of facts and certainties will never reach it. And that means that a system that leans on facts (science, our legal system, philosophy) needs to add a certain component that is not just another fact, to establish some kind of consistency that could pass for truth in order to derive conclusions and verdicts. Science uses the phrase 'most likely' to seal the gap. Law uses the phrase 'beyond reasonable doubt'. In both cases an appeal is made to the trust of the spectators. Trust is needed in any kind of theory in order to breach the gap caused by the inherent incompleteness of any system that uses facts for its bricks.
Where o where does trust come from? The human heart, perhaps? Intuition? Not by a long shot. As Jeremiah once wrote, "The heart is the most deceitful thing."
When Koch came out with his snow flake, mathematicians found it horrendous as it went directly against all intuition that a finite surface could be encompassed by an infinite periphery. But it did. Weirdness ruled. The earth wasn't flat after all.
Still, theoretical physicists had already surmised of something similar that most likely actually occurs in nature. Due to extreme spacetime curvature, a black hole has an infinite radius and a finite diameter. Hence volume is infinite while circumference is far from it. How is this possible? It just is. Intuition is most often flat wrong. And what about the universe itself? It is the only thing in existence with an inside and no outside...
The new buzz-word in theoretical physics these days is String Theory. One of the things this young and pretty theory suggests is that besides the regular four dimensions of space-time there are a cool seven extra dimensions of pure weirdness wrapped up into extremely tiny balls of dimensional yarn.
Go figure, and while you're at it, go to the next chapter:
The Most Deceitful Thing