Monday, May 2, 2011

The beauty of Incompleteness Theorem of Kurt Gödel

Mathematics and Physics played a major role in our humanity to observer and understand  the world we live in and they helped a lot in filling the gaps and answered the age old questions we humans curiously looked for with observations and proven tests.
Most of us take for granted that Maths is the absolute realization of physics, its like physics is the blue print and maths is the concrete/cement or the language to be precise.
But, the naked fact about maths is the entire theory of numbers was built on assumptions and axioms that they should be absolute and rational, when we look into the history of mathematics, mathematicians were not comfortable in considering irrational numbers and imaginary numbers which obviously plays a vital role in current dynamics for building a system of theory based on axioms with tangible results, ironically the then mindset of scientists and mathematicians who tried to ignore them completely realized the important properties of numbers that are irrationality and imaginary. The whole branch of Quantum Mechanics deals with them. Mathematicians and Scientists from several fields acknowledged and realized that they can never ignore the nature of numbers, which put an end to the old mindset of scientific research. Well, at least, gladly that's how a rational mind works, we wouldn't hesitate to accustom to the truth by leaving the prejudice behind. This may not be even related to the context of this post. But felt like saying. 
Here is an interesting research done by one of the pioneers in logical analysis and a renowned mathematician.
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. 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.
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.
  • A logical system is always based on axioms.
  • A logical system can not prove the consistency of its own axioms and can hence not prove whether itself is true.
  • No logical system will ever be able to prove everything.
  • Truth can not be reached by logic.
  • Truth is singular (Truth is One).
And here is my Conclusion
Actually, nothing. I am a rationalist at heart, for me anything that works or makes sense in a rationalistic point of view is always the authentic and credible source, of course, my opinion may contradict with this post, but that's what I have chosen to follow. Nevertheless, the whole idea of Gödel works is interesting and thought provoking to be honest.

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