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Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.

Nice find. I quickly glanced through the articles and it seems that the cosmic fingerprints guy's real target should be Hume, not Godel. And we've come a long way since Hume..

Any system of logic has to be based on axioms and on some accepted rules for making statements for 'logical' reasoning. Axioms are 'truths' we take as givens. There can be bad axioms and good axioms. Here is a bad axiom: 'God creates and runs everything.' It is bad for at least two reasons: (i) The term 'God' cannot be defined in a self-consistent and logical manner. (ii) Although this axiom 'explains' everything, we really end up understanding nothing.

Gregory Chaitin has argued at length that what Gödel's theorem has done is to make mathematicians concede that mathematics HAS to be done more and more like physics. In physics we accept (at least for the time being) certain things as facts (axioms) if there is overwhelming empirical evidence to support them, even though not all 'facts' have the sanctity of mathematical proof. Mathematicians believe in proving their theorems. Chaitin argues that it is sensible to accept certain 'conjectures' and 'postulates' in mathematics as theorems (use some other word if you do not want to call them theorems), because their proofs may never become available. I fail to understand where 'God' enters that argument, whatever the term 'God' means.

I have gone through the article by Perry Marshall. Here is my response to his key statements.

1. 'Faith and Reason are not enemies. In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.'

What is the big deal? In mathematics we have axioms. The selection of axioms is made on some rational basis. They are not just a matter of blind FAITH.


2. 'Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect.'

There is a logical contradiction in your statement. If you have indeed drawn the biggest circle, then there can be nothing outside it. You will have to go on drawing bigger and bigger circles, ad infinitum. You are not allowed to stop somewhere!


3. 'In the history of the universe we also see the introduction of information, some 3.8 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.'

This is an absurd statement. Perry knows nothing about the meaning of information, let alone the subject of information theory.


4. 'The information had to come from the outside, since information is not known to be an inherent property of matter, energy, space or time.'

This is an absurd statement again. Perry knows nothing about the meaning of information, let alone the subject of information theory.


5. 'All codes we know the origin of are designed by conscious beings.'

Juvenile stuff.


6. 'Therefore whatever is outside the largest circle is a conscious being.'

There is a logical contradiction in your statement. If you have indeed drawn the biggest circle, then there can be nothing outside it. You will have to go on drawing bigger and bigger circles, ad infinitum. You are not allowed to stop somewhere!


7. 'The Incompleteness of the universe isn’t proof that God exists. But… it IS proof that in order to construct a consistent model of the universe, belief in God is not just 100% logical… it’s necessary.'

Science surely does not have all the answers. But the scientific method is the only method which can give us progressively better answers. 'Belief in God is necessary' is like saying 'It is necessary to give a name to our present ignorance about certain matters'.

I think the concept of observability in control theory is relevant to this post.

The main implication is this:
If we do not know a priori what the system equation is and we have access only to the outputs of the system then irrespective of for how long we measure the outputs we can give a description of only the observable part of the system.

If we now consider the system to be the universe as a whole (of course with a lot of hand-waving) then the implication is that we can only hope to ever give a description of the "observable" part of the universe. And, one could say that god constitutes the unobservable part (whose existence cannot be ruled out).

Any thoughts? Am I making some mistake?

(27-05-2011 08:25 AM)P11 Wrote:  And, one could say that god constitutes the unobservable part (whose existence cannot be ruled out).

Please try defining the term ‘God’ in your statement in a precise and self-consistent manner. It will be interesting to see how you that!

(27-05-2011 10:32 PM)Vinod Wadhawan Wrote:  

(27-05-2011 08:25 AM)P11 Wrote:  And, one could say that god constitutes the unobservable part (whose existence cannot be ruled out).

Please try defining the term ‘God’ in your statement in a precise and self-consistent manner. It will be interesting to see how you that!

The caveat of a congruence theory of truth, which follows from Godel's incompleteness theorem, is that it relies on the validity of concepts with respect to the axiomatic concepts in the framework in which they are framed, in other words, if you have two axiomatically diametrically opposed, but self-consistent and empirically untestable systems, they are both equally valid, firstly, and secondly they say nothing about whether the axioms themselves have ontological representation (i.e, whether they actually exist)

What this means is that while deities can exist in some axiomatic deductive systems, it would be erroneous to claim that these deities definitely have ontological existence, or that these axioms represent absolute reality (the existence of which itself has to be assumed axiomatically, nudge nudge wink wink)

The reason it is erroneous to claim this is that an axiom can either be true or false insofar ontological correspondence is concerned, but if you assert that it definitely is or is not by taking a leap of faith, you are arguing that a possibly empty set definitely contains an element, which is logically absurd. (A set with an element cannot be empty by definition)

So, if someone claims that some versions of deities that are logically possible actually exist one can just point out using the same leap of faith in the other direction that they don't exist. To sum it all up, the flaw in the argument is going from a may exist ------> a definitely exists, which would also justify going from a may not exist ---------------> a definitely does not exist.

(01-10-2010 09:00 PM)Lije Wrote:  Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.

Despite all the wordiness of the theorem it is quite simple, if you are using a non-absurd axiomatic deductive system which is powerful enough to be able to describe addition of elements, you cannot prove those axioms using the same system, you may however derive those from the axioms of a more complex system, and the axioms of that system from an even more complex system and so on.

In other words, you cannot prove axioms using the same logical system that they are a part of unless they are sufficiently simple, and only first order logic would fit the bill here, where true = true, false = false and true =/= false.

This is why, unless empirical testing is possible, one cannot declare one non-absurd self consistent system as superior or truer than another. You can argue about different frameworks of ethics all you want, but you cannot say one is truer than the other because you cannot prove those axioms in the same framework. Absurd systems, on the other hand, are completely useless since they allow contradictions to be true, and therefore everything to be true.

(01-10-2010 09:00 PM)Lije Wrote:  Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.

The theorem basically implies that in a "Consistent " system - that is a system which is based on a few axioms such that anything can then be derived from it contains statements relevant to the system which can neither be proved or disproved using the said axioms.
Perhaps the biggest fallacy in applying godel in real life is that we have not derived our Universe out of a few axioms, ie , proved that it is a consistent system.

(18-09-2011 09:49 PM)Alan DSouza Wrote:  

(01-10-2010 09:00 PM)Lije Wrote:  Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.

The theorem basically implies that in a "Consistent " system - that is a system which is based on a few axioms such that anything can then be derived from it contains statements relevant to the system which can neither be proved or disproved using the said axioms.
Perhaps the biggest fallacy in applying godel in real life is that we have not derived our Universe out of a few axioms, ie , proved that it is a consistent system.

Alan, that will be far bigger blow to science than anything if it turns out that our universe is not consistent. By the way I thought that Godel also proved that any sufficiently complicated axiomatic system cannot prove its consistency.

In any case the article seems to take the implications of logic literally physically. There is nothing outside or inside about logic which the author interprets as entity outside the universe etc. Don't know what such a fallacy is called.

(19-09-2011 03:13 PM)Kanad Kanhere Wrote:  In any case the article seems to take the implications of logic literally physically. There is nothing outside or inside about logic which the author interprets as entity outside the universe etc. Don't know what such a fallacy is called.

This can be considered an instance of the fallacy of reification where 'logic', an abstract construct is treated as a physical entity.

Misunderstanding recursion is a prerequisite for mystic interpretations of mathematics and computer science.

Gödel's novel trick of assigning numbers to mathematical statements is a matter of expedience, just like Euclid's use of reductio ad absurdum 2000 years before Gödel. It is exactly the results of this kind of intellectual effort that make almost every mathematical proof, however insignificant, beautiful to those who love mathematics.

This is yet another mathematical concept that has needlessly been mystified. Others include, prime numbers, imaginary numbers, the numbers (±√5 +1 )/2 , chaotic dynamics, etc. I am just waiting for the day when someone mystifies something abstruse like Baire category theorem, http://en.wikipedia.org/wiki/Baire_category_theorem . It won't obviously happen, because it. Just. Cannot. Be. Explained. In. 'Layman's'. 'Terms'.

[Edit: fixed typos.]