Are there mathematical premises that don't hold true whithin other mathematical systems/ways of thinking? I do not really know how to phrase the question, but it is as follows: Are there mathematical premises that don't hold truth given different systems? And can someone give me examples?
For example take: the inside angles of a triangle, in Euclidean space, sum up to 180 degrees. However it is not possible to demonstrate that the interior angles of a triangle equal 180 degrees in Euclidean space, within other systems, such as spherical geometry or hyperbolic geometry.
 A: In standard (“classical”) logic, if you can prove that a statement is true, you can conclude that it is not false, and if you can prove it is not false, you can conclude that it is true. In intuitionistic logic, the first is acceptable, but the second one is considered invalid.
(In notation, intuitionistic logic accepts $P\to\lnot\lnot P$ but rejects $\lnot\lnot P\to P$.  “$x\to y$” means that from $x$ you can infer $y$, and “$\lnot x$” means that $x$ is false.)
A: To put some perspective on the example you gave, imagine a world in which we knew about the counting numbers $1,2\ldots,$ but we didn't know about $0.$ We might try, as the Greeks did with geometry, to write down some axioms of arithmetic: $$x +y =y+x\\x+(y+z)=(x+y)+z\\xy=yx\\x(yz)=(xy)z\\x(y+z)=xy+xz.$$ We might think it reasonable that we should be able to prove the well-known fact that $x+y\ne x,$ but try as we might, we would never be able to. That's because we could be working in the naturals $\{0,1,2\ldots\},$ instead of the counting numbers $\{1,2,\ldots\},$ in which case our axioms still hold, but the "well-known fact" is not true, so the fact cannot be a consequence of these axioms.
(Actually, the axioms I wrote down are even weaker than that: they don't even preclude the case of a very boring world in which zero is the only number.)
So examples like the case of the parallel postulate that you mention are pretty easy to come by. For instance, the idea that "every polynomial has a root" is absurd with respect to the real numbers, but is true in the complex numbers.
