How to know when all the facts are in

The Judge wrote

I guess it just goes to show how people can sometimes think something is true without having all the facts to hand.

How can one know when one has all the facts (without assuming that you have all the facts)? If this is possible, then I'd love to know the criteria. If it is impossible, then all knowledge is tentative.

Having given this about ten seconds of thought, I would say that one usually cannot say that all the facts have been accounted for, except perhaps in cases of pure mathematics, i.e. 2 + 2 = 4, the sum of the internal angles of a traingle is 180 degress (in Euclidean geometry, of course), etc.

[Philboid Studge]

ocmpoma wrote

Having given this about ten seconds of thought, I would say that one usually cannot say that all the facts have been accounted for, except perhaps in cases of pure mathematics, i.e. 2 + 2 = 4, the sum of the internal angles of a traingle is 180 degress (in Euclidean geometry, of course), etc.

This is slightly out of my depth (by 'slightly,' I mean 'way'), but didn't Gödel show there could be cracks of doubt even in pure math?

From Boyer's History of Mathematics:
'Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.'

I see from Wikipedia that Gödel's first theorem is 'one of the most misunderstood,' and I'm probably proving them right. Any math people want to weigh in?

Philboid Studge wrote

I see from Wikipedia that Gödel's first theorem is 'one of the most misunderstood,'

Yeah, Godel's theorem seems like one of those things where someone with a little knowledge and an agenda can seriously mis-represent what the theorem's all about. Kind of like quantum mechanics, and the corresponding "quantum flapdoodle" that results.

Not to say that I have a clue what that theorem's all about, aside from what Joe Average knows.

Basically, the theorem states that in any complex system of mathematics, there are some correct (ture) statements that cannot be proven to be correct using that system. To make it simpler - he proved that there is no complete system of mathematics. Which is something along the lines of why I specifically mentioned Euclidean geometry when talking about triangles.

[Rhinoqulous]

ocmpoma wrote

Basically, the theorem states that in any complex system of mathematics, there are some correct (ture) statements that cannot be proven to be correct using that system. To make it simpler - he proved that there is no complete system of mathematics. Which is something along the lines of why I specifically mentioned Euclidean geometry when talking about triangles.

Which is why math is different from logic. Logic is a complete system, math is not. I studied logic, and learned about this, but I know many physics and math geeks who still claim that math and logic are one and the same thing.

Rhinoq