Gödel's incompleteness theorems (moved to Freethinking Anonymous)
Posted on: August 25, 2007 - 3:50pm
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Gödel's incompleteness theorems (moved to Freethinking Anonymous)
Gödel says that a system cannot be complete AND consistent at the same time. Two of my friends use this thereom as an argument that science cannot explain everything and therefore something supernatural can/must exist. I cannot follow their argumentations, but I have to admit that reading Gödel's theorems gives me a headache, so that I usually cannot give any strong counterarguments.
Are there any experts here, who can enlighten me on this subject, so that I'm prepared the next time this discussion comes up? Thank you in advance.
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Gödel says that any formal system is not complete or not consistent. The various sciences are not formal systems.
Furthermore, a formal system being incomplete doesn't mean something can't be explained. It means there's a true statement in the system that has no formal proof.
[Edit: grammar]
Yeah, we have been there before, but they argue that formal systems (e.g. arithmetic) provide the syntax of science.
That's good enough for me, but not for them.
Using mathematics doesn't make a branch of science a formal system. To my knowledge, no branch of science has been reduced to a formal system. Have your friends done this? If so, they should publish the results, and probably plan a trip to Norway to get their Nobel Prize. No branch of science uses formal, deductive proofs, which disqualifies any branch of science from being a formal system. Gödel's theorem simply doesn't apply.
Formal systems are in the domain of logic and mathematics, not science. Most mathematicians don't even bother with formal proofs, and Gödel's theorem has never hindered any any mathematician.
What I've stated is the definition of a complete formal system. There's no "not good enough." It just is. If your friends don't want to use this definition, they can't use Gödel's theorem. Period.
Let's assume that we can, and have, reduced biology to a formal system. Let's say we want to settle whether or not the mind is a function of the brain. So somehow we have this statement, "The mind is a function of the brain", in this formal system for biology.
This statement is either true or false. Ideally, you would then use the rules of inference of the system, along with the axioms and any other useful results, to provide a formal, deductive proof of the statement's truth or falsehood.
Now, Gödel's theorem says that either our formal system for biology is incomplete or inconsistent. Let's assume it's consistent, so we have an incomplete formal system. This means that there exists a statement that is true that has no proof. But, does this mean that the statement "The mind is not a function of the brain." is true and has no proof? No. The theorem only tells us of the existence of such statements. It doesn't tell us what the statements are. On the flip side, "The mind is a function of the brain." could be true (i.e. humans are complete physical), but have no proof.
The point? Even if we could apply Gödel's theorem, it doesn't support your friend's supernaturalism.
Thank you very much for this explanation. Now I know what to say the next time we talk atheism vs. theism.