Wednesday, 20 February 2008

Wayward thoughts on Gödel's incompleteness theorem

As the Kurt Gödel's incompleteness theorem got me going, I felt that I have to look at it more thoroughly. The thoughts brought forward in the Miskatonic University Press website by William Denton, give an account which I feel it would direct my thoughts.

It is mentioned that:

"In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements."

As my objective is to use Gödel's theorem beyond the discipline it is meant for, namely mathematics, to include all disciplines, and not all. It is evident, even in the Miskatonic University Press website, that the implications far exceed the narrow boundaries of mathematics, and extends into all structures built by the human kind. Even the individual itself is under the command of the incompleteness theorem. I wonder whether that is a result of the limitations of the human mind, as is incapable of being anything other than a simplified mind, and it affects us in more ways than we could imagine. The further implications are that all systems built in humanity's tenacious attempts to understand reality as well as the application of these constructed systems in organising our social structures bears the hallmark of our imperfect consciousnesses. Which it might be attributed to the structure of our brains.

And this tendency affects all systems built, no matter how awesome, elaborate or elegant they might appear to be, to us. Certainly nobody can deny that humanity has achieved great things in its passage through life, therefore the acceptance of imperfect consciousnesses is invalidated. However it betrays a constant yearning towards even more achievements never to be satisfied with what has previously been accomplished. The doubt in-built within the human mind carries it along even further.

What it reveals is the futility of rigidly adhering to rules and other forms of prescriptive norms since these should constantly be questioned and modified in the same way as in mathematics, where you might be able to prove every conceivable statement about numbers within a system, by going outside the system, in order to come up with new rules and axioms, regardless whether you'll only create a larger system with its own unprovable statements. Evolution will take care of the new larger system that has been created.

Evolution will rule and provide the direction that a given system will take and adjust the rules which will apply for the new larger system. Constitutions and laws that govern relationships of individuals in societies bear witness as a brief look at history shows. As in the first societies the laws started to protect the rights of a king or a monarch and gradually they were engineered to protect the rights of even more and more individuals in a society to the state they are now. Are the laws in modern societies engineered to protect all individuals amidst them? Continued unrest and dissatisfaction in all societies globally, says not. The larger system, the global world, demands new rules and axioms and to that drive nobody can go against, no matter what they do. The force is exponentially strengthened as more and more individuals are involved.

Systems will crumple and their fall will reflect in intensity to the resistance they put out. Inversely proportional of how hard they fight to stall such process. Going against the free exchange of thoughts in whatever way imagined, is the same as going against the rules and axioms our larger global world demand for, what evolution will eventually bring forth.

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