Friday, July 20, 2007
The Constraint of Godel
Godel was 20th century mathematician who in my view propunded one of the most important scientific theories of all time. Why i am singling him out is because what he said about 'knowability'. His view showed us that there are limits which any theoritical system's provability. Though in this google-and-wiki world we can all know what he said by typing a few keystrokes, it is very important to understand the implication of what he said.
His essential point was : Any mathematical system cannot be fully provable, complete and consistent on their own at the same time. THe essential point to grasp is that Godel is not saying that this is because of any flaw of our logical sophistication. He is essentially making a point that some things are "unknowable" despite the best logic.
He 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. One 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. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. This is an important point - the fact that somethings are 'fundamentally' unknowable. No amount of human effort or rationality is going to take this away.
It is often said that Godel made an important point to self-referencibility. Any universal system is falliable as it cannot explain itself. To explain itself it has to be 'outside' it and in that sense it is 'incomplete' (since it cannot explain itself).
Though it is perhaps a spurious comparison, i am reminded here of Quantum mechanics. The Hiesburg uncertainty principle which states that one cannot find the position and the momentum of an fundamental particle specifically. One of the two will remain unknowable.
One of the most important implications i believe is about self-awarness. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself. Essentially you cannot step outside your skin to view yourself from 'outside'
PS : A great book to explain is the EGB (Ester, Godel, Bach - the eternal golden Braid by Douglas Hofstader)
His essential point was : Any mathematical system cannot be fully provable, complete and consistent on their own at the same time. THe essential point to grasp is that Godel is not saying that this is because of any flaw of our logical sophistication. He is essentially making a point that some things are "unknowable" despite the best logic.
He 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. One 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. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. This is an important point - the fact that somethings are 'fundamentally' unknowable. No amount of human effort or rationality is going to take this away.
It is often said that Godel made an important point to self-referencibility. Any universal system is falliable as it cannot explain itself. To explain itself it has to be 'outside' it and in that sense it is 'incomplete' (since it cannot explain itself).
Though it is perhaps a spurious comparison, i am reminded here of Quantum mechanics. The Hiesburg uncertainty principle which states that one cannot find the position and the momentum of an fundamental particle specifically. One of the two will remain unknowable.
One of the most important implications i believe is about self-awarness. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself. Essentially you cannot step outside your skin to view yourself from 'outside'
PS : A great book to explain is the EGB (Ester, Godel, Bach - the eternal golden Braid by Douglas Hofstader)
# posted by Abhijat @ 8:24 PM 
