Comment Re:Terrible, Terrible, Headline (Score 1) 154
You cannot prove that parallel lines do not intersect purely by using geometry. You cannot prove 1+1=2 using math. The former is treated as an axiom, a statement that is intuitively assumed to be true in further proofs. The latter is a definition of terms - given what 1, 2, addition and equality are defined as, the statement is true.
The former is not treated as an axiom it is an axiom and as such it cannot be proven in any system. The latter can indeed be proven if you step outside of the bounds of algebra (which is a field of mathematics that covers addition) and into the bounds of set theory where we have a definition for 1 and a definition for 2 and a definition for the process of addition and using those three definitions we can indeed prove that 1 + 1 = 2. In algebra we would define the addition of our system such that 1 + 1 = 2 at which point it is treated as an axiom. To be clear there are systems where 1 + 1 != 2 (boolean algebra) but we can assume you knew that and were specifically discussing integer arithmetic.