Posted: May 26, 2019 7:31 pm
aufbahrung wrote:My background on mathematics is purely functional. I never understood the idea of a proof. Seems like symbolic tautology.
It is. Although I suspect from the way you use those words you might not use them in their mathematical sense.
aufbahrung wrote:Can someone explain why a proof ain't saying the same thing with 'different words' for me?
A proof is a sequence of steps that gets from a set of axioms or assumptions to a conclusion which is not identical with any any of those axioms or assumptions.
Desireable conclusions are often ones which whilst being much narrower than the set of tautological consequences of the axioms of your theory are much more directly applicable. E.g. Cantor's diagonal argument showing that there are different infinite cardinalities for the reals and naturals.
One of the key concepts is that if something is proven then it is true, but if it's not proven (or in some cases even provable) then you don't know if it's true.
aufbahrung wrote:How can a proof be valid?
By the conclusion following from the axioms (or premises) by a sequence of steps all of which follow the rules of the deductive system you're working in.
aufbahrung wrote:I know equations work and everything...but isn't it a monstrous waste of time and effort to proove the obvious?
Mostly you don't prove obvious things. For example Wiles's Theorem (Fermat's last theorem) follows from the axioms of number theory, but it was not obvious for hundreds of years that it was actually a true statement.