Godel Begets God ?
I am not a Mathematician (though I have absence of a social life, a necessary condition to be a mathematician), but aren't Godel's theorems a very strong arguement for validity of concept of God ? Why don't we hear of it ?
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16 comments:
No, they only say that some statements can be neither proved nor disproved by so-and-so logic.
You might find this interesting.
I'm convinced that those who try too hard to bridge the physical and the spiritual are doomed to insanity.
Those coming from the side of spirituality end up with results that are not replicated.
Those coming from the side of empiricism keep adding new codicils and tweaks, but like Zeno never quite catch the truth. (And like Pirsig in "Zen and the Art of Motorcycle Maintenance," literally go mad.)
Those who deny value in the other side are just petty and not worth exploring life with.
The reason Godel is neither cited nor refuted is because most people aren't smart enough to even understand just what the hell Godel proved. The philosophical interpretations usually tend to be unintelligent bastardizations, the way post modernist philosophers misused the theory of relativity to 'establish' the 'relativity of truth' and suchlike.
Godel is probably my favourite philosopher/mathematician. The reason I refrain from mentioning him anywhere is because I fear I myself will end up bastardizing his theorems.
Ritwik,
I agree understanding Godel's theorem requires decent technical knowledge. But it is not the case that people comment only on things they understand - I am not talking just about the "popularity" of string theory but also the way society works, for example.
Yup, Godel's enough to check most scientists who are focused on proving religious absurdity. Thanks, I mentioned him in my new post.
Our rationalists ought to divert their scientific energy towards the strength of the archeological evidence collected by the ASI. (We are told they did not even study the area).
How scientific is that ?
Frog,
Thanks, what I actually meant was doesnt Godel actually do away with the belief that is possible to have a perfectly logical system (or something like that)
Ike,
Yes, perhaps a better way is to maintain harmony between spiritual and empirical.
Ritwik,
No harm in trying is there ?
RC,
It has nothing to with science it is more to do with the naraative.
Godel's theorem doesn't prove/disprove the existance of God, instead it disproves most of the existing notions of God (To completely understand the system.. you have to be atleast on the circumfrence of the circle which describes the system...since god is outside the circle..all, if not most of the existing notions are wrong)
PS:Godels theorem has already been misused by those un-intelligent design un-funnies to un-make their un-point.
Gaurav,
On a personal note,i was dissappointed .What has God got to do with Godel.They are two different ways of making sense of life.
The word "God" might mean different things to different people.I think you should have been more precise.
In formal systems,like arithmetic,a statement could be true or false.Some might be just undecidable.
The existence of God(its truth or falsity could be one such within the ordinary meaning of language and science).Karl Popper sees scientific theories as something essentially not provable to be true,but can be shown to be false.This is in the context of science.
Even in science theories are "fruitful" when they provide a structure/perspective but the test of a theory is correlation with experiment.(all well known)
In mathematics,these investigation has stemmed from two sources.One whether Euclid's fifth postulate is consistent with other axioms.It was discovered that this is independent of other axioms.(non-euclidean geometry)So the truth/falsity of this axiom is not absolute.
Godels work was inspired by russel's work to reduce mathematics to logic. In the foundations of mathematics,among mathematicians,the formalists(followers of the great German mathematician David Hilbert have always had greater support,ie,if they have been inclined to think on the foundation.The other school is the intuitionist(constructivist school of the dutch mathematician,Brouwer).The professional mathematician's viewpoint is well articulated by Courant in his famous work"what is mathematics".
Godels famous incompleteness theorem showed that even in simple formal systems there are undecidable statements.This put paid all the hopes of Russel.
ie one cannot decide the truth/falsity of these statements within the system.
Most mathematicians are working in particular problems/theories.They think a particular problem can be solved one way or the other.(proof or counter-example)
Yet the mathematics community were stunned(it was so reported) when Stanley Cohen(a Stanford scholar) proved the axiom of choice is independent of other axioms.This is an undecidable statement in the settheory(and it has an important implication for real numbers.The surprise is because the real numbers are such a familiar(and complex) object.A recent attempt(i found in the links www.rutgers.math,you had referred,dismisses the notion of continuity,infinite and tries to build the entire edifice on discreteness and finiteness.Whatever we are discussing,would be then meaningless)
One of Hilbert's original 23 problems(given in the 1900 international mathematical congress),a problem on the possibility of a certain algorithm in number theory,turned out be undecidable.(proved by the russian margulis).This again stunned mathematicians.(generally proof or counter example).
Most great scientific acievements(Newton,Einstein) or the genius of Ramanujan in number theory spring from the depths of human intellect which are always a source of awe/mystery to the ordinary man.With effort and if we are sufficiently curious we can "know"what these minds have discoverd.
Godel's theorems shows the limitations of logic even in elementary formal system.The grandiose plan of russel to reduce mathematics to logic failed.(Ofcourse,i am not denigrating russel's search,for they were the primary inspiration of godel's research,nor was russel a simpleton.As a tripos scholar,he was aware of the richness of mathematics)
In that sense,Godels work shows the limitation of logic and our awe/mystery of the world remains undimmed.
PS:Godel like other scientists had the greatest respect for the pursuit of knowledge and order.
xyz and others,
To put it in extremely non-formal language - the Godel's incompletenes theorem is this :
An axiomatic logical system cannot be both consistent (no contradictions) and complete (all true statements can be derived from base axioms) at the same time. Hence, a logically consistent system will allow certain statements that are known to be true, but are unprovable within the axioms of that system.
The extension w.r.t God is - A logical system that is based on certain axioms (which can be the axioms that we build upon physics on - the individisibilty of the Planck's constant or the speed of light, for instance)and that seeks to describe the universe will make certain statements about the universe (the existence of God, or a transcendent entity)which it will not be able to prove. The lack of proof does not imply the falsehod of such an assertion. Hence, it could be perfectly rationally consistent to claim that 'God exists'. This statement should then be debated on the merits of its consistency with the logical system that is chosen, and not considered false only because it cannot be conclusively proven given the axioms of our system.
In short, it is possible to be rational, scientific, and a believer at the same time without any contradictions or any extraneous compulsions to prove the existence of God as long as one is extremely smart with the choice of the axioms that one thinks defines the universe.
Godel did not imply the existence of God. He only implied that it may not be irrational to believe in God and science at the same time.
There are assumptions in Godel's theorems. One is that the axioms should have some correspondence with the natural numbers and should be extendable. This means that extending Godel to the universe will mean taking an essentially computational view of the universe (computational, not determinstic - there's a difference). I'm not too sure of the philosophical implications of the assumptions.
It is true that Godel's theorems were targeted to end the trend of logical formalism in mathematics. However, they do have supra-mathematical extensions.
Zen,
i was writing in a garbled way from what i had read in the past.
How do you know that certain statements are true but not provable within the axioms of the system.
I am not asking for a proof from mathematical logic.Are there examples from mathematics(natural numbers,groups ) or other systems wherein we know a statement to be true,but it cannot be proved within the axioms or is the proof an existence proof of such statements.
zen,
did i make any serious mistakes in what i wrote.It was anything but lucid,but did i make any serious error.regarding the examples,i gave.
Zen,
Do you that such an exercise as defining God within the axioms of physics and trying to define its consistency will have any human interest?
This might seem a very narrow question.But then we choose to pursue whatever that interests us.What would be the motive in defining such a concept of God.You are precocious,would you be interested in doing such an exercise if you had the time.
We find that Leibniz,Desacartes and pascal gave 'persuasive' arguments for the existence of God,either because of their own belefs or the by the spirit of timee.Unless a person has deep faith in God(which has little to do with logic),why should he indulge in such an exercise?
xyz,
No there were no serious mistakes. I just wrote with reference to your'what does Godel have to do with God' question.
Godel's proof of incompleteness is a reductio ad absurdum, or proof by contradiction. If one assumes that a logical system is complete, it turns out to be inconsistent. Hence, a consistent system must definitely be incomplete.
An example of a statement that is true but unprovable is - (A and A')' = 1 (true). Any 'proof' of this statement will assume the statement itself, and as such is circular. Of ourse this is a trivial example - an expert in mathematical logic will be able to provide better ones.
As for the attempt to formalize the notion of God in terms of axioms, it is concurrent with the aim to formalize the universe in terms of certain basic axiomatic principles. Godel's theorem implies that the Grand Unified Theory may still have space for god. Godel was himself a believer, but he has not provided any existence proofs. he has only formally codified Leibnitz's ontological proof. The Cartesian and Pascal-ian proofs have long been refuted.
As for me, I wouldn't be interested in such exercises myself, but then I'm not researching the GUT either. Any attempts will probably be made by people who are believers themselves - you are right on that count. However, when the theism/atheism debate is of such intllectual importance to several people, it would be wrong to assume that such a attempt will be of little human interest. In any case, for the purpose of a debate, the human interest in the attempt is irrelevant. The only thing that matters is the correctness of the logic and its consistency within the system.
I'm off to watch the Pakistan innings now. I doubt if I can do any further justice to the debate - I've pretty much exhausted all I had to say. Thanks.
Zen,
Thanks.Let me have one final say.
1)Perhaps you can post on Godel's theorem.You have the ability to explain so very well.Perhaps,you can take the examples(Euclidean/non euclidean and the axiom of choice) for consistency and Hilberts problem for completeness.
2)math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf has an exposition on ultrafinitism.Philosophically it does not appeal to me.Perhaps,you can take a look.
3)On the philosophical motivation for defining God through certain axioms,why should any believer take the trouble of defining God to be merely consistent with axioms.Would it not be more reasonable to expect that God emerges from the axioms to be the prime mover at the heart of the universe?
4)Secondly any theory of God should be have some "explanation" for the creation and the state of the universe as it is now.The ancient philosophers did not have to contend with the wealth of information in diverse fields ranging from geology to astronomy.How feasible would such a project be?I am not ofcourse discounting the existence of very clever minds or who knows even Divine inspiration?But atleast for some one like me,it looks so remote.
4)On a related subject,perhaps our ancient seers(who confronted bewildering diversity)had no final pronouncements to make on Iswara.
As an expiation for some of my postings( i feel guilty),i am posting this
"That I,with bitter sweating brow
No more may I teach what I do not know;
That I with piercing ken may see
The world's indwelling energy;
The hidden seeds of life explore
And deal in empty words and forms no more.
-Goethe(Faust-opening scene).
This is my confession.It is not against anyone or any intellectual enquiry.
Interesting discussion even though I didn't understand anything education has its uses :-(
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