What are the philosophical implications of Godel's incompleteness theorem?

TheEyeThatEats

As I understand it, Godel's theory shows that any formal mathematical or logical system will always contain things that are true that cannot be proved within the agreed axioms of that system.

Does this imply then that by the process of reason there will be things that are conceptually invisible, or "intuitively" true even though one cannot prove it?

Or am I misunderstanding?

Priori

It seems to me that the most obvious things of all are not only impossible to prove, but must ground proof itself.

Reason must always start somewhere, with something 'given'.

Morbert

TheEyeThatEats wrote: »

As I understand it, Godel's theory shows that any formal mathematical or logical system will always contain things that are true that cannot be proved within the agreed axioms of that system.

Not quite.

http://en.wikipedia.org/wiki/Self-verifying_theories

Morbert

Priori wrote: »

It seems to me that the most obvious things of all are not only impossible to prove, but must ground proof itself.

Reason must always start somewhere, with something 'given'.

This is true. Even Solipsim, or "I think therefore I am", have unprovable axioms. Logic and reason is all about consistency, not necessary truth.

The physicist Roger Penrose compares reasoning to Plato's world of forms. We establish theorems which necessarily follow from axioms, but no compulsion to adopt axioms.

Grayson

Godel believed that you could separate logic into rtwo types. Formal and intuitive. And they could at times contradict each other. Formal logic has always been the logic that rationalists cite. Which makes perfect sense. But Godel believed that formal logic wasn't comprehensive. He believed that there were times when it wouldn't work.

After he completed the incompletness theorm, he was asked to explain it in layman terms. He said it was like constructing a series of laws for a country. If you tried to make laws that covered every single eventuality, you'd end up with a situation where eventually you would find laws that contradict each other. It was better to limit the amount of laws and have use intuitive logic to figure out how the laws should be applied in particular situations. In reality, thet's why we have judges. They are (supposed to be ) smart people knowledgable in the law with the experience to make educated decisions about how the law should be applied.

Does that make sense?