Originally posted by MITCHEL
Theorists claim it ultimately is not possible to prove anything...
Whether you're aware of it or not, you've hit the worst nightmare of formal logic and mathematics in general -
Gödel's incompleteness
theorems.
It actually goes a little bit different than what you've stated here.
It's not that you cannot prove anything, it's that there will always be something that you cannot prove. The proof itself depends on the set of
axioms (facts taken for granted), wihich can be consistent, or inconsistent. With inconsistent set you can prove anything, so that's not very useful.
With consistent set you're stuck with unprovable theorems.
In simple words - what you choose to belive in in the first place, determines what you can prove and what you can't. But (and there's the kicker in
the mathematical a**), even if you choose an infinite set of axioms, you will still end up with things that you just can't prove.
What Gödel showed is that in most cases, such as in number theory or real analysis, you can never create a complete and consistent finite
list of axioms, or even an infinite list that can be produced by a computer program. Each time you add a statement as an axiom, there will always be
other true statements that still cannot be proved as true, even with the new axiom. Furthermore if the system can prove that it is consistent, then it
is inconsistent.
It is possible to have a complete and consistent list of axioms that cannot be produced by a computer program (that is, the list is not
computably enumerable).
(/qdv7y)
While Wikipedia's reference is mostly for those interested in mathematics, the following site explains it in simpler terms.
In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some
propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to
prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by
doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by
definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of
rules.
(
www.miskatonic.org...)
Anyway, those are really interesting theorems. Especially considering that science, which is 100% based on mathematics, is usually thought to (some
day) explain (or describe) all of the Reality.
Science may actually be able to "explain" "something" some day, but it will never be able to
prove those "explanations"... I believe
that's what you were aiming at.
This actually makes science just a little bit better than religion which is 100% based on faith (or sets of axioms, mathematically speaking; simply
facts taken for granted, remember?).
The only advantage of science over religion is - experiment. The rest is all the same, only in different packages.
Once again I come to the same conclusion -
the only thing that matters is personal experience.
Everything else is based on axioms... which someone, somewhere, told us to just take as "Truths".
[edit on 26-3-2008 by elendal]